Brilliant Integration Contest - Season 1

I am interested in holding an Integration Contest here on Brilliant.org like any other online forums such as AoPS or Integrals and Series. The aims of the Integration Contest are to improve skills in the computation of integrals, to learn from each other as much as possible, and of course to have fun. Anyone here may participate in this contest.

The rules are as follows

  1. I will start by posting the first problem. If there is a user solves it, then (s)he must post a new one.
  2. You may only post a solution of the problem below the thread of problem and post your proposed problem in a new thread. Put them separately.
  3. Please make a substantial comment.
  4. Make sure you know how to solve your own problem before posting it in case there is no one can answer it within a week, then you must post the solution and you have a right to post another problem.
  5. If the one who solves the last problem does not post his/her own problem after solving it within a day, then the one who has a right to post a problem is the last solver before him/her.
  6. The scope of questions is only computation of integrals either definite or indefinite integrals.
  7. You are NOT allowed to post a multiple integrals problem as well as a complex integral problem.
  8. You are also NOT allowed to post a solution using a contour integration or residue method.
  9. The final answer can ONLY contain the following special functions: gamma function, beta function, Riemann zeta function, Dirichlet eta function, dilogarithm, digamma function, and trigonometric integral.

Please post your solution and your proposed problem in a single new thread.

Format your post is as follows:

SOLUTION OF PROBLEM xxx (number of problem) :

[Post your solution here]

PROBLEM xxx (number of problem) :

[Post your problem here]

Please share this note so that lots of users here know this contest and take part in it. (>‿◠)✌

Okay, let the contest begin! Here is the first problem:

PROBLEM 1 :

For a>0a>0, show that

0alnxaxx2dx=πln(a4) \int_0^a \frac{\ln x}{\sqrt{ax-x^2}}\,dx=\pi\ln\left(\frac{a}{4}\right)

P.S. You may also want to see Brilliant Integration Contest - Season 1 (Part 2) and Brilliant Integration Contest - Season 1 (Part 3).

#Calculus #Integration #IntegrationTechniques #Contests #Integrals

Note by Anastasiya Romanova
6 years, 6 months ago

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Here is the solution: I=0aln xaxx2dx\displaystyle I = \int \limits_0^a \frac{ \text{ln } x }{\sqrt{ax - x^2}} \text{d}x

I=0aln (ax)axx2dx\displaystyle I = \int \limits_0^a \frac{ \text{ln } (a-x) }{\sqrt{ax - x^2}} \text{d}x

2I=0aln (axx2)axx2dx\displaystyle \Rightarrow 2I = \int \limits_0^a \frac{ \text{ln } (ax - x^2) }{\sqrt{ax - x^2}} \text{d}x

Put x=a2(1sint)axx2=a24cos2t\displaystyle x = \frac{a}{2} (1 - \sin t ) \Rightarrow ax - x^2 = \frac{a^2}{4} \cos ^2 t

Therefore, the integral becomes, 2I=π2π2ln (a24cos2t)a24cos2t(a2cost)(dt)\displaystyle 2I = \int \limits_{\frac{\pi}{2}}^{\frac{-\pi}{2}} \frac{ \text{ln } (\frac{a^2}{4} \cos ^2 t ) }{\sqrt{\frac{a^2}{4} \cos ^2 t }} (\frac{a}{2} \cos t )(-\text{d}t)

Rearranging, we obtain,
2I=40π2ln (a2cost)dt\displaystyle 2I = 4\int \limits_0^{\frac{\pi}{2}} \text{ln } (\frac{a}{2} \cos t ) \text{d}t

The value of 0π2ln (cosθ)dθ\displaystyle \int \limits_0^{\frac{\pi}{2}} \text{ln } ( \cos \theta ) \text{d}\theta is π ln 22\displaystyle \frac{ -\pi \text{ ln } 2 }{2} (which I have calculated separately and I can post if it is required).

Thus, we obtain, I=π ln (a2)2π ln 22=π ln (a4)\displaystyle I = \pi \text{ ln } \bigg( \frac{a}{2} \bigg) - 2\frac{ \pi \text{ ln } 2 }{2} = \pi \text{ ln } \bigg(\frac{a}{4} \bigg)

Sudeep Salgia - 6 years, 6 months ago

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Find a closed form expression for the integral : I=sin(2015x)sin2013x dx\displaystyle I = \int \sin (2015 x) \sin ^{2013} x \text{ d}x

Sudeep Salgia - 6 years, 6 months ago

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Solutionofproblem2 Solution \quad of \quad problem \quad 2

Split sin(2015x)sin(2015x) as sin(2014x+x)sin(2014x+x) and our integral becomes :

I=(sin2014x.cos(2014x)+sin(2014x)cos(x)sin2013(x))dxI=\displaystyle \int { ({ sin }^{ 2014 }x.cos(2014x)+sin(2014x)cos(x){ sin }^{ 2013 }(x))\quad dx }

Multiply and divide it by 20142014 to get :

12014(sin2014x.(2014cos(2014x))+sin(2014x)(2014sin2013(x)cos(x)))dx\frac { 1 }{ 2014 } \displaystyle \int { ({ sin }^{ 2014 }x.(2014cos(2014x))+sin(2014x)(2014{ sin }^{ 2013 }(x)cos(x)))\quad dx }

I=12014sin2014(x)dsin(2014x)+sin(2014x)(dsin2014x)\Rightarrow \displaystyle I=\frac { 1 }{ 2014 } \int { { sin }^{ 2014 }(x)dsin(2014x)+sin(2014x)(d{ sin }^{ 2014 }x) }

I=12014d(sin2014(x)sin(2014x))\Rightarrow \displaystyle I=\frac { 1 }{ 2014 } \int { d({ sin }^{ 2014 }(x)sin(2014x)) }

I=sin2014(x)sin(2014x)2014+C\large \displaystyle I=\frac { { sin }^{ 2014 }(x)sin(2014x) }{ 2014 } +C

Problem3\large Problem \quad 3

Evaluate I=0(sin(x)x)2dxI=\displaystyle \int _{ 0 }^{ \infty }{ { (\frac { sin(x) }{ x } })^{ 2 }dx }

Ronak Agarwal - 6 years, 6 months ago

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@Ronak Agarwal Perfect. Nice work Ronak. Expecting a problem soon.

Sudeep Salgia - 6 years, 6 months ago

@Ronak Agarwal Nicely done, +1. Now you must post a new problem, but before that. I think it would be nice if you post this as a new post, no need to reply @Sudeep Salgia. See the rules & the format post. Thank you. :)

Anastasiya Romanova - 6 years, 6 months ago

@Ronak Agarwal Where is your proposed problem? We are waiting it. Please don't let us wait too long like this. You should post your solution and your proposed problem at once.

Anastasiya Romanova - 6 years, 6 months ago

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@Anastasiya Romanova @Anastasiya Romanova I have posted my problem sorry for holding the contest.

Ronak Agarwal - 6 years, 6 months ago

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@Ronak Agarwal It's okay. I have posted the solution of your problem and also proposed a new problem. :)

Anastasiya Romanova - 6 years, 6 months ago

Nicely done, +1! But you forget to post your problem anyway. You have a privilege for that so that this contest keeps going on. Post PROBLEM 2 in your thread solution. Thanks :)

Anastasiya Romanova - 6 years, 6 months ago

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Well, there must be a rule for people like Ronak! Solving a problem doesn't give you the right to withhold the contest. In such a case, someone should be allowed to interfere, right?

Satyam Bhardwaj - 6 years, 6 months ago

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@Satyam Bhardwaj Sorry If that troubled all because actually I have to go to coaching centre, and my internet connection got down and I went to my coaching centre, I have returned just now and has posted the question.

Ronak Agarwal - 6 years, 6 months ago

@Satyam Bhardwaj OK, if no-one posts his/her own problem after solving a problem within a day. The one who has a right to post a problem is the last solver before him/her. Is this fair?

Anastasiya Romanova - 6 years, 6 months ago

Solution of Problem 3 :

Use integration by parts by taking u=sin2xu=\sin^2x and dv=dxx2dv=\dfrac{dx}{x^2}, we get

0sin2xx2dx=sin2xx0+02sinxcosxxdx=0+0sin2xxdx=0sinttdtt=2x\begin{aligned} \int_0^\infty\frac{\sin^2x}{x^2}\,dx&=-\left.\frac{\sin^2x}{x}\right|_0^\infty+\int_0^\infty\frac{2\sin x\cos x}{x}\,dx\\ &=0+\int_0^\infty\frac{\sin 2x}{x}\,dx\\ &=\int_0^\infty\frac{\sin t}{t}\,dt\qquad\Rightarrow\qquad t=2x\\ \end{aligned}

Now consider

I(a)=0eatsinttdta0 I(a)=\int_0^\infty\frac{e^{-at}\sin t}{t}\,dt\qquad a\ge0

so that I()=0I(\infty)=0 and our considered integral is I(0)I(0). Differentiating w.r.t. aa and then integrating back, we get

I(a)=0eatsintdt=11+a2integration by parts twiceI(a)=11+a2da=arctan(a)+C\begin{aligned} I'(a)&=-\int_0^\infty e^{-at}\sin t\,dt\\ &=-\frac{1}{1+a^2}\qquad\Rightarrow\qquad\text{integration by parts twice}\\ I(a)&=-\int \frac{1}{1+a^2}\,da\\ &=-\arctan(a)+C \end{aligned}

For I()=0I(\infty)=0, implying C=π2C=\dfrac{\pi}{2}. Hence

I(a)=0eatsinttdt=π2arctan(a) I(a)=\int_0^\infty\frac{e^{-at}\sin t}{t}\,dt=\frac{\pi}{2}-\arctan(a)

and

I(0)=0sinttdt=π2 I(0)=\int_0^\infty\frac{\sin t}{t}\,dt=\frac{\pi}{2}

Thus

0sin2xx2dx=π2 \int_0^\infty\frac{\sin^2x}{x^2}\,dx=\frac{\pi}{2}

Problem 4 :

Prove

0π/2ln(cosx)sinxdx=π28 \int_0^{\pi/2}\frac{\ln(\cos x)}{\sin x}\,dx=-\frac{\pi^2}{8}

Anastasiya Romanova - 6 years, 6 months ago

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SolutionofProblem4\large Solution\quad of\quad Problem\quad 4

Firstly we will prove a general result

Result

01xaln(x)dx=1(1+a)2\displaystyle \int _{ 0 }^{ 1 }{ { x }^{ a }ln(x)dx } =\frac { -1 }{ { (1+a) }^{ 2 } }

Proof

I=01xaln(x)dxI=\displaystyle \int _{ 0 }^{ 1 }{ { x }^{ a }ln(x)dx }

Integrating by parts u=ln(x),dv=xadxu=ln(x),dv={x}^{a}dx

I=ln(x)xa+1a+1011a+101xadxI=\displaystyle ln(x)\frac { { x }^{ a+1 } }{ a+1 } \overset { 1 }{ \underset { 0 }{ | } } -\frac { 1 }{ a+1 } \int _{ 0 }^{ 1 }{ { x }^{ a }dx }

I=1(1+a)2I=\frac { -1 }{ { (1+a) }^{ 2 } }

Hence proved

Now we have :

I=0π2ln(cos(x))sin(x)dxI=\displaystyle \int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { ln(cos(x)) }{ sin(x) } dx }

Take cos(x)=ycos(x)=y to get our integral as :

01ln(y)dy1y2\displaystyle \int _{ 0 }^{ 1 }{ \frac { ln(y)dy }{ 1-{ y }^{ 2 } } }

Now 11y2=n=0y2n\frac { 1 }{ 1-{ y }^{ 2 } } =\displaystyle \sum _{ n=0 }^{ \infty }{ { y }^{ 2n } }

Our integral becomes :

I=n=001y2nln(y)dyI=\displaystyle \sum _{ n=0 }^{ \infty }{ \int _{ 0 }^{ 1 }{ { y }^{ 2n }ln(y)dy } }

Using our proved result we get :

I=n=01(2n+1)2I=-\displaystyle \sum _{ n=0 }^{ \infty }{ \frac { 1 }{ { (2n+1) }^{ 2 } } }

Also the summation can we written as :

I=(n=11n2n=11(2n)2)I=-(\displaystyle \sum _{ n=1 }^{ \infty }{ \frac { 1 }{ { n }^{ 2 } } } -\sum _{ n=1 }^{ \infty }{ \frac { 1 }{ { (2n) }^{ 2 } } } )

I=(ζ(2)14ζ(2))=34ζ(2)=π28I=-(\zeta (2)-\frac { 1 }{ 4 } \zeta (2))=\frac { -3 }{ 4 } \zeta (2)=\large \frac{-{\pi}^{2}}{8}

Where ζ(x)\zeta(x) is the zeta function

Problem5\large Problem \quad 5

Find closed form of I=0dx1+xn\displaystyle I=\int _{ 0 }^{ \infty }{ \frac { dx }{ 1+{ x }^{ n } } }

Ronak Agarwal - 6 years, 6 months ago

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Keep going fella, +1! But please post your solution & proposed problem in a new thread like I did. No need to reply other threads to post a solution & a problem. Anyway, I've posted the solution of your problem & also proposed a new problem.

Anastasiya Romanova - 6 years, 6 months ago

Solution of Problem 7:

Proposition :

0πdxp+cosxdx=πp21\begin{aligned} \int_0^{\pi} \frac{dx}{p+\cos x}\,dx=\frac{\pi}{\sqrt{p^2-1}} \end{aligned}

Proof :

It can be proven by using Weierstrass substitution: t=tan(x2)t=\tan\left(\dfrac{x}{2}\right), then

0πdxp+cosxdx=02p+1+(p1)t2dtt=p+1p1tany=2p21arctant  0=πp21Q.E.D.\begin{aligned} \\ \int_0^{\pi} \frac{dx}{p+\cos x}\,dx &=\int_0^{\infty} \frac{2}{p+1+(p-1)t^2}\,dt\qquad\Rightarrow\qquad t=\sqrt{\frac{p+1}{p-1}}\tan y\\ &=\left.\frac{2}{\sqrt{p^2-1}}\arctan t\;\right|_0^{\infty}\\ &=\frac{\pi}{\sqrt{p^2-1}}\qquad\qquad\text{Q.E.D.} \end{aligned}

Differentiating the proposition w.r.t. pp twice and setting p=10p=\sqrt{10}, we have

2p20π1p+cosxdx=2p2[πp21]0π2(p+cosx)3dx=π(2p2+1)(p21)50π1(10+cosx)3dx=7π162\begin{aligned} \\ \frac{\partial^2}{\partial p^2}\int_0^\pi\frac{1}{p+\cos x}\, dx&=\frac{\partial^2}{\partial p^2}\left[\frac{\pi}{\sqrt{p^2-1}}\right]\\ \int_0^\pi\frac{2}{\left(p+\cos x\right)^3}\, dx&=\frac{\pi\left(2p^2+1\right)}{\sqrt{\left(p^2-1\right)^5}}\\ \int_0^\pi\frac{1}{\left(\sqrt{10}+\cos x\right)^3}\, dx&=\frac{7\pi}{162} \end{aligned}

Problem 8 :

Prove

0π2dx1+8sin2(tanx)=π6(2e2+12e21) \int_0^{\Large\frac{\pi}{2}}\frac{dx}{1+8\sin^2(\tan x)}=\frac{\pi}{6}\left(\frac{2e^2+1}{2e^2-1}\right)

Anastasiya Romanova - 6 years, 6 months ago

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Solution of Problem 8:

Substitute tanxx\tan x\mapsto x, then the integral is:

0dx(1+x2)(1+8sin2x)=0dx(1+x2)(54cos(2x))\int_0^{\infty} \frac{dx}{(1+x^2)(1+8\sin^2x)}=\int_0^{\infty} \frac{dx}{(1+x^2)(5-4\cos(2x))}

=011+x2(13(1+2k=1cos(2kx)2k))=π6+23k=112k0cos(2kx)1+x2dx=\int_0^{\infty} \frac{1}{1+x^2}\left(\frac{1}{3}\left(1+2\sum_{k=1}^{\infty} \frac{\cos(2kx)}{2^k}\right)\right)=\frac{\pi}{6}+\frac{2}{3}\sum_{k=1}^{\infty} \frac{1}{2^k}\int_0^{\infty} \frac{\cos(2kx)}{1+x^2}\,dx

=π6+π3k=1(12e2)k=π6+π312e21=π6(2e2+12e21)=\frac{\pi}{6}+\frac{\pi}{3}\sum_{k=1}^{\infty} \left(\frac{1}{2e^2}\right)^k=\frac{\pi}{6}+\frac{\pi}{3}\frac{1}{2e^2-1}=\boxed{\dfrac{\pi}{6}\left(\dfrac{2e^2+1}{2e^2-1}\right)}

I have used the following result:

0cos(mx)x2+a2dx=π2aeam\int_0^{\infty} \frac{\cos(mx)}{x^2+a^2}\,dx=\frac{\pi}{2a}e^{-am}

I cannot think of a challenging problem at the moment, I request somebody else to post one. Thanks!

Pranav Arora - 6 years, 6 months ago

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Ingenious! But I have a doubt - In your fourth step, how is it guaranteed that the function cos(2kx)2k\dfrac{\cos (2kx)}{2^k} is positive on (0,)(0,\infty)? PS : you forgot to write the dx\mathrm{d}x in the third step.

Pratik Shastri - 6 years, 6 months ago

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@Pratik Shastri I don't know what did you mean by positive/negative on (0,)(0,\infty), but if you meant to swap between integral & summation sign, it is valid because cos(2kx)1+x2\frac{\cos(2kx)}{1+x^2} is continuous, finite & integrable, therefore swapping those two signs can be justified by Fubini's theorem.

Anastasiya Romanova - 6 years, 6 months ago

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@Anastasiya Romanova If pranav refuses to put up a question then who put will put it up @Anastasiya Romanova .

Ronak Agarwal - 6 years, 6 months ago

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@Ronak Agarwal According to the rule 4, the one is the last solver and it turns out that one is me. See PROBLEM 9 below, I've just proposed it.

Anastasiya Romanova - 6 years, 6 months ago

@Anastasiya Romanova Right. I confused it with Tonelli's theorem.

Pratik Shastri - 6 years, 6 months ago

Sorry for very late response. Using elementary techniques,-

Solution - Problem 7

0πdxpcosx \displaystyle \int_{0}^{\pi} \dfrac{dx}{p - cosx}

2I=0π2p dxp2cos2x 2I = \displaystyle \int_{0}^{\pi} \dfrac{2p~dx}{p^2 - cos^2x}

I=0πp dxp2(1+tan2x)1 I = \displaystyle \int_{0}^{\pi} \dfrac{p~dx}{p^2(1 + tan^2x) - 1}

I=p0πsec2x dxp2tan2x+(p21)2 I = p \displaystyle \int_{0}^{\pi} \dfrac{sec^2x~dx}{p^2tan^2x + \sqrt{(p^2 - 1)^2}}

tanx = t

I=2p0dtp2t2+(p21)2 I = 2p \displaystyle \int_{0}^{\infty} \dfrac{dt}{p^2t^2 + \sqrt{(p^2 - 1)^2}}

I=2pp21×1p[0tan1ptp21 I = \dfrac{2p}{ \sqrt{p^2 - 1}} \times \dfrac{1}{p} \Big[_{0}^{\infty} tan^{-1} \dfrac{pt}{ \sqrt{p^2 - 1}}

I=πp21 I = \dfrac{\pi}{ \sqrt{p^2 - 1}}

U Z - 6 years, 5 months ago

Solution to problem 10

First integrate by parts -

I=0sin3xx2dx=03sin2xcosxxdxI= \int_{0}^{\infty} \dfrac{\sin^3{x}}{x^2} dx =\int_{0}^{\infty} \dfrac{3\sin^2{x} \cos{x}}{x} dx

Then, use the property of the laplace transform that L{f(t)t}(s)=sF(p)dp\mathcal{L} \left\{\dfrac{f(t)}{t}\right\}(s)=\displaystyle\int_{s}^{\infty} F(p) dp (in our case, s0s \rightarrow 0).

I=30L{sin2tcost}(p) dpI=3\int_{0}^{\infty} \mathcal{L} \{\sin^2{t} \cos{t}\}(p) \ dp

I=302s(s2+1)(s2+9)ds=3log34I=3\int_{0}^{\infty} \dfrac{2s}{(s^2+1)(s^2+9)} ds=\boxed{\dfrac{3\log {3}}{4}} The last integral can be found by using partial fractions.

Note : L{f(t)}(s)=0estf(t)dt\mathcal{L}\{f(t)\}(s)=\displaystyle\int_{0}^{\infty} e^{-st} f(t) \mathrm{d}t

Problem 11

Find

0logxx2+α2dx     for α>0\int_{0}^{\infty} \dfrac{\log {x}}{x^2+\alpha^2} \mathrm{d}x \ \ \ \ \ \text{for} \ \alpha>0

Pratik Shastri - 6 years, 6 months ago

Solution of Problem 13 :

First we prove the following proposition:

Proposition :

01xalnkx dx=(1)kk!(a+1)k+1for  k=0,1,2,\int_0^1 x^a \ln^k x\ dx=\frac{(-1)^k\, k!}{(a+1)^{k+1}} \qquad\text{for }\ k=0,1,2,\ldots

Proof :

Note that 01xa dx=1a+1for  α>1. \int_0^1 x^a\ dx=\frac1{a+1}\qquad\text{for }\ \alpha>-1. Differentiating equation above kk times w.r.t. aa we have 01kak(xa) dx=01xalnkx dx=(1)kk!(a+1)k+1Q.E.D. \int_0^1 \frac{\partial^k}{\partial a^k}\left(x^a\right)\ dx=\int_0^1 x^a \ln^k x\ dx=\frac{(-1)^k\, k!}{(a+1)^{k+1}}\qquad\text{Q.E.D.}

Now, we will evaluate the general case of

0xa1ebx1dx=0xa1ebx1ebxdxfor  a,b>0\int_0^\infty\frac{x^{a-1}}{e^{bx}-1}\,dx=\int_0^\infty\frac{x^{a-1}\,e^{-bx}}{1-e^{-bx}}\,dx\qquad\text{for }\ a,b>0

Set y=ebxy=e^{-bx}, then

0xa1ebx1dx=(1)a1ba01lna1y1ydy\int_0^\infty\frac{x^{a-1}}{e^{bx}-1}\,dx=\frac{(-1)^{a-1}}{b^a}\int_0^1\frac{\ln^{a-1}y}{1-y}\,dy

Use a geometric series for 11y\dfrac{1}{1-y} then interchange the integral and summation sign which is justified by the Fubini–Tonelli theorem and apply the previous proposition, we have

0xa1ebx1dx=(1)a1ba01k=0yklna1ydy=(1)a1bak=001yklna1ydy=(1)a1bak=0(1)a1(a1)!(k+1)a=(a1)!bak=11ka=Γ(a)ζ(a)ba\begin{aligned} \int_0^\infty\frac{x^{a-1}}{e^{bx}-1}\,dx&=\frac{(-1)^{a-1}}{b^a}\int_0^1\sum_{k=0}^\infty y^k\,\ln^{a-1}y\,dy\\ &=\frac{(-1)^{a-1}}{b^a}\sum_{k=0}^\infty\int_0^1 y^k\,\ln^{a-1}y\,dy\\ &=\frac{(-1)^{a-1}}{b^a}\sum_{k=0}^\infty \frac{(-1)^{a-1}\, (a-1)!}{(k+1)^{a}}\\ &=\frac{(a-1)!}{b^a}\sum_{k=1}^\infty \frac{1}{k^{a}}\\ &=\frac{\Gamma(a)\,\zeta(a)}{b^a} \end{aligned}

where Γ(a)\Gamma(a) is the gamma function and ζ(a)\zeta(a) is the Riemann zeta function.

Hence, by setting a=n+1a=n+1 and b=1b=1, we have

0xnex1dx=Γ(n+1)ζ(n+1)\int_0^\infty\frac{x^{n}}{e^{x}-1}\,dx=\Gamma(n+1)\,\zeta(n+1)

Problem 14 :

Prove that

01ln(3+x3x)dxx(1x)=πln(7+435+26)\begin{aligned} \int_{0}^1 \ln\left(\frac{3+x}{3-x}\right)\,\frac{dx}{\sqrt{x(1-x)}}=\pi\ln\left(\dfrac{7+4\sqrt{3}}{5+2\sqrt{6}}\right)\\ \end{aligned}

Anastasiya Romanova - 6 years, 6 months ago

Since @Pranav Arora is unable to propose a problem (I hope it's only temporary) and to make this contest sustains, then according to rule 4, I, as the last solver, have a right to propose a new one. Here is the problem:

PROBLEM 9

Prove that

01ln(1+x1x)dxx1x2=π22\int_0^1 \ln\left(\frac{1+x}{1-x}\right)\frac{dx}{x\sqrt{1-x^2}}=\frac{\pi^2}{2}

Anastasiya Romanova - 6 years, 6 months ago

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Solution -Problem - 9 We substitute, t=1+x1x    x=1t1+t t = \frac{1+x}{1-x} \implies x = \frac{1-t}{1+t} Doing the substitution and , simplifying , gives, I=01ln(t)(t)(1t)dt I = - \int_{0}^{1} \frac{\ln(t)}{(\sqrt{t})(1-t)} dt Now we substitute, t=sin2(x) t=\sin^2(x) I=40π2ln(sin(x))cos(x)dx I = -4\int_{0}^{\frac{\pi}{2}} \frac{\ln(\sin(x))}{\cos(x)} dx Using, Problem -44 I=4×π28=π22 I = -4 \times \frac{-\pi^2}{8} = \frac{\pi^2}{2}

Shivang Jindal - 6 years, 6 months ago

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I had already posted the solution to the problem and a new problem also you should not reply instead post your solution as a seperate comment as I did @Shivang Jindal

Ronak Agarwal - 6 years, 6 months ago

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@Ronak Agarwal Lol, time difference of 5mins. I think i was typing solution when you posted the solution .

Shivang Jindal - 6 years, 6 months ago

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@Shivang Jindal You can try my posted question. @Shivang Jindal

Ronak Agarwal - 6 years, 6 months ago

I am able to write the integral as a series - I=πn=0(2n)!22n(n!)2(2n+1)I=\pi \sum_{n=0}^{\infty} \dfrac{(2n)!}{2^{2n}(n!)^2(2n+1)}

Pratik Shastri - 6 years, 6 months ago

Solutionofproblem9Solution\quad of\quad problem\quad 9

I=01ln(1+x1x)dxx1x2\displaystyle I=\int _{ 0 }^{ 1 }{ ln(\frac { 1+x }{ 1-x } )\frac { dx }{ x\sqrt { 1-{ x }^{ 2 } } } }

Put x=cos(θ)x=cos(\theta) to get our integral as :

0π2ln(cot2θ2)dθcosθ\displaystyle \int _{ 0 }^{ \frac { \pi }{ 2 } }{ ln({ cot }^{ 2 }\frac { \theta }{ 2 } )\frac { d\theta }{ cos\theta } }

I=(2)0π2ln(tanθ2)dθcosθ\Rightarrow \displaystyle I=(-2)\int _{ 0 }^{ \frac { \pi }{ 2 } }{ ln(tan\frac { \theta }{ 2 } )\frac { d\theta }{ cos\theta } }

Put tan(θ/2)=xtan(\theta/2)=x to get our integral as :

(4)01ln(t)dt1t2\displaystyle (-4)\int _{ 0 }^{ 1 }{ \frac { ln(t)dt }{ 1-{ t }^{ 2 } } }

I proved in my solution to problem 4 that :

01ln(t)dt1t2=π28\displaystyle \int _{ 0 }^{ 1 }{ \frac { ln(t)dt }{ 1-{ t }^{ 2 } } } =\frac { -{ \pi }^{ 2 } }{ 8 }

Using this I get :

I=π22I=\frac{{\pi}^{2}}{2}

Problem10Problem\quad 10

Find 0sin3xx2dx\displaystyle \int _{ 0 }^{ \infty }{ \frac { { sin }^{ 3 }x }{ { x }^{ 2 } } dx }

Ronak Agarwal - 6 years, 6 months ago

Solution of Problem 11 :

Using substitution u=α2x    x=α2u    dx=α2u2 duu=\dfrac{\alpha^2}{x}\;\Rightarrow\;x=\dfrac{\alpha^2}{u}\;\Rightarrow\;dx=-\dfrac{\alpha^2}{u^2}\ du, then

0lnxx2+α2 dx=0ln(α2u)(α2u)2+α2α2u2 duI(α)=02lnαlnuα2+u2 du=2lnα01α2+u2 du0lnuu2+α2 du=2lnα01α2+u2 duI(α)I(α)=lnα01α2+u2 du.\begin{aligned} \int_0^\infty\frac{\ln x}{x^2+\alpha^2}\ dx&=\int_0^\infty\frac{\ln \left(\dfrac{\alpha^2}{u}\right)}{\left(\dfrac{\alpha^2}{u}\right)^2+\alpha^2}\cdot \dfrac{\alpha^2}{u^2}\ du\\ I(\alpha)&=\int_0^\infty\frac{2\ln \alpha-\ln u}{\alpha^2+u^2}\ du\\ &=2\ln \alpha\int_0^\infty\frac{1}{\alpha^2+u^2}\ du-\int_0^\infty\frac{\ln u}{u^2+\alpha^2}\ du\\ &=2\ln \alpha\int_0^\infty\frac{1}{\alpha^2+u^2}\ du-I(\alpha)\\ I(\alpha)&=\ln \alpha\int_0^\infty\frac{1}{\alpha^2+u^2}\ du. \end{aligned}

The last integral can easily be evaluated by using substitution u=αtanθu=\alpha\tan\theta, then

0lnxx2+α2 dx=lnαα0π2 dθ=πlnα2α\begin{aligned} \int_0^\infty\frac{\ln x}{x^2+\alpha^2}\ dx&=\frac{\ln \alpha}{\alpha}\int_0^{\Large\frac\pi2} \ d\theta\\ &=\frac{\pi\ln \alpha}{2\alpha} \end{aligned}

Problem 12 :

Prove

cos(arctan2x)(1+x2)1+4x2dx=π3 \int_{-\infty}^{\infty}\frac{\cos \left(\, \arctan 2x\right)}{(1+x^2)\sqrt{1+4x^2}}\,dx=\frac{\pi}{3}

Anastasiya Romanova - 6 years, 6 months ago

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The last substitution should be u=αtanθu=\alpha \tan \theta :)

Pratik Shastri - 6 years, 6 months ago

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Fixed it. Sorry, I'm too hasty. Thanks... :)

Anastasiya Romanova - 6 years, 6 months ago

This is a simple kind of NCERT problem. It can be easily done by substituting t=tan1xα t = tan^{-1}\dfrac{x}{\alpha} and then applying by parts

U Z - 6 years, 5 months ago

SolutionofProblem12Solution\quad of\quad Problem\quad 12

It's a very good disguised integral.

First note that the function is an even function hence our integral can be written as :

I=20cos(arctan(2x))(1+x2)1+4x2dx\displaystyle I=2\int _{ 0 }^{ \infty }{ \frac { cos(arctan(2x)) }{ (1+{ x }^{ 2 })\sqrt { 1+4{ x }^{ 2 } } } dx }

Now we know that cos(arctan(2x))=11+4x2\displaystyle cos(arctan(2x))=\frac{1}{\sqrt{1+4{x}^{2}}}. Using this our integral becomes :

I=201(1+x2)(1+4x2)dx\displaystyle I=2\int _{ 0 }^{ \infty }{ \frac { 1 }{ (1+{ x }^{ 2 })(1+4{ x }^{ 2 }) } dx }

Splliting it by partial fractions we get :

I=23(0dx(14+x2)0dx1+x2)\displaystyle I=\frac { 2 }{ 3 } (\int _{ 0 }^{ \infty }{ \frac { dx }{ (\frac { 1 }{ 4 } +{ x }^{ 2 }) } } -\int _{ 0 }^{ \infty }{ \frac { dx }{ 1+{ x }^{ 2 } } } )

I=23(2tan1(2x)0tan1(x)0)\displaystyle I=\frac { 2 }{ 3 } (2{ tan }^{ -1 }(2x)\overset { \infty }{ \underset { 0 }{ | } } -{ tan }^{ -1 }(x)\overset { \infty }{ \underset { 0 }{ | } } )

I=π3\displaystyle I=\frac { \pi }{ 3 }

Problem13Problem\quad 13

Find closed form of I=0xnex1dx\displaystyle I=\int _{ 0 }^{ \infty }{ \frac { { x }^{ n } }{ { e }^{ x }-1 } dx }

Ronak Agarwal - 6 years, 6 months ago

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Wait a sec!? Could you rectify your solution, because the original problem is written as arctan(2x)\arctan(2x). And one thing, what did you mean by "Now we know that cos(arctanx)=11+4x2\cos(\arctan x)=\frac{1}{\sqrt{1+4x^2}}"? Could you elaborate?

Anastasiya Romanova - 6 years, 6 months ago

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That was my typing mistake, also Letarctan(2x)=θ arctan(2x)=\theta, hence 2x=tan(θ)2x=tan(\theta), hence cos(θ)=11+4x2cos(\theta)=\frac{1}{\sqrt{1+4{x}^{2}}}

cos(arctan(2x))=11+4x2\Rightarrow cos(arctan(2x))=\frac{1}{\sqrt{1+4{x}^{2}}}

Ronak Agarwal - 6 years, 6 months ago

Solutionofproblem14Solution\quad of\quad problem\quad 14

Lemma

π2π2ln(a2cos2x+b2sin2x)dx=2πln(a+b2)\displaystyle\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ ln({ a }^{ 2 }{ cos }^{ 2 }x+{ b }^{ 2 }{ sin }^{ 2 }x)dx } =2\pi ln(\frac { a+b }{ 2 } )

Proof :

I(a)=π2π2ln(a2cos2x+b2sin2x)dx\displaystyle I(a)=\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ ln({ a }^{ 2 }{ cos }^{ 2 }x+{ b }^{ 2 }{ sin }^{ 2 }x)dx }

Differentiating with respect to aa we get :

I(a)=2aπ2π2cos2xa2cos2x+b2sin2xdx\displaystyle {I}^{'}(a)=2a\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \frac { { cos }^{ 2 }x }{ { a }^{ 2 }{ cos }^{ 2 }x+{ b }^{ 2 }{ sin }^{ 2 }x } dx }

Put tan(x)=ttan(x)=t to get our integral as :

I(a)=2aπ2π2dt(a2+b2t2)(1+t2)\displaystyle {I}^{'}(a)=2a\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \frac { dt }{ ({ a }^{ 2 }+{ b }^{ 2 }{ t }^{ 2 })(1+{ t }^{ 2 }) } }

=2aπ2π2dt(a2+b2t2)(1+t2)=2ab2a2(π2π2b2dta2+b2t2π2π2dt1+t2)\displaystyle =2a\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \frac { dt }{ ({ a }^{ 2 }+{ b }^{ 2 }{ t }^{ 2 })(1+{ t }^{ 2 }) } } =\frac { 2a }{ { b }^{ 2 }-{ a }^{ 2 } } (\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \frac { { b }^{ 2 }dt }{ { a }^{ 2 }+{ b }^{ 2 }{ t }^{ 2 } } } -\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \frac { dt }{ 1+{ t }^{ 2 } } } )

Solving it we get :

I(a)=2πa+b\displaystyle {I}^{'}(a)=\frac { 2\pi }{ a+b }

I(a)=2πln(a+b)+C\Rightarrow \displaystyle I(a)=2\pi ln(a+b)+C

Put a=b=1a=b=1 to get C=2πln(2)C=-2\pi ln(2)

Hence I(a)=2πln(a+b2)\displaystyle I(a)=2\pi ln(\frac{a+b}{2})

In our integral in the given question put x=1sinθ2x=\frac{1-sin\theta }{2} to get the integral as :

I=π2π2ln(7sin(θ))dxπ2π2ln(5+sin(θ))dxI= \displaystyle \int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ ln(7-sin(\theta))dx } -\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ ln(5+sin(\theta))dx }

Using the property abf(x)dx=abf(a+bx)dx\displaystyle \int _{ a }^{ b }{ f(x)dx } =\int _{ a }^{ b }{ f(a+b-x)dx } we get :

I=π2π2ln(7+sin(θ))dxπ2π2ln(5sin(θ))dx\displaystyle I=\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ ln(7+sin(\theta))dx } -\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ ln(5-sin(\theta))dx }

Adding these two we get :

I=12(π2π2ln(49sin2θ)dxπ2π2ln(25sin2(θ))dx)\displaystyle I=\frac { 1 }{ 2 } (\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ ln(49-{ sin }^{ 2 }\theta)dx } -\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ ln(25-{ sin }^{ 2 }(\theta))dx } )

I=12(π2π2ln(49cos2θ+48sin2θ)dxπ2π2ln(25cos2θ+24sin2θ)dx)\displaystyle I=\frac { 1 }{ 2 } (\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ ln(49{ cos }^{ 2 }\theta+48{ sin }^{ 2 }\theta)dx } -\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ ln(25{ cos }^{ 2 }\theta+24{ sin }^{ 2 }\theta)dx } )

Using our given lemma we get :

I=12(2πln(7+432)2πln(5+262))=πln(7+435+26)\displaystyle I=\frac { 1 }{ 2 } (2\pi ln(\frac{7+4\sqrt { 3 }}{2} )-2\pi ln(\frac{5+2\sqrt { 6 }}{2} ))=\pi ln(\frac { 7+4\sqrt { 3 } }{ 5+2\sqrt { 6 } } )

Problem15Problem\quad 15

Evaluate I=011ln(x)+11xdx\displaystyle I=\int _{ 0 }^{ 1 }{ \frac { 1 }{ ln(x) } +\frac { 1 }{ 1-x } dx }

Ronak Agarwal - 6 years, 6 months ago

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According to timeline, you're faster than @jatin yadav to post your solution, then the right to propose a new problem is yours.

Anastasiya Romanova - 6 years, 6 months ago

_ Solution of problem 14 _

I=01ln(3+x3x)1x(1x2)dxI = \displaystyle \int_{0}^{1} \ln \bigg(\dfrac{3+x}{3-x} \bigg) \dfrac{1}{\sqrt{x(1-x^2)}} {\mathrm dx}

= 01ln(1+x/3)1x(1x)dx\displaystyle \int_{0}^{1} \ln(1 + x/3) \dfrac{1}{\sqrt{x(1-x)}} {\mathrm dx}

= 0π/22ln(1+13sin2θ)dθ0π/22ln(113sin2θ)dθ\displaystyle \int_{0}^{\pi/2} 2 \ln\bigg(1 + \frac{1}{3} \sin^2 \theta\bigg) {\mathrm d} \theta - \displaystyle \int_{0}^{\pi/2} 2 \ln\bigg(1 - \frac{1}{3} \sin^2 \theta\bigg) {\mathrm d} \theta

Lemma 0π/2ln(1+asin2θ)=πln(1+a+12)\displaystyle \int_{0}^{\pi/2} \ln(1+a \sin^2 \theta) = \pi \ln \bigg(\dfrac{1+\sqrt{a+1}}{2}\bigg)

Proof

Let I(a)=0π/2ln(1+asin2x)dxI(a) = \displaystyle \int_{0}^{\pi/2} \ln(1+a \sin^2 x) {\mathrm dx}

I(a)=0π/2sin2x1+asin2xdxI'(a) = \displaystyle \int_{0}^{\pi/2} \dfrac{\sin^2 x}{1+ a \sin^2 x} {\mathrm dx}

I(a)=0π/21a(1sec2x1+(a+1)tan2xdx)I'(a) = \displaystyle \int_{0}^{\pi/2} \dfrac{1}{a}\bigg(1 - \dfrac{\sec^2 x}{1+(a+1) \tan^2x} {\mathrm dx}\bigg)

= π2a1a(0dz1+(a+1)z2) \dfrac{\pi}{2a} - \dfrac{1}{a} \bigg(\displaystyle \int_{0}^{\infty} \dfrac{dz}{1+(a+1)z^2}\bigg)

=π2a(11a+1)\dfrac{\pi}{2a} \bigg(1 - \dfrac{1}{\sqrt{a+1}}\bigg)

Hence, I(a)I(0)=0aπ2(1x1xx+1)dxI(a) - I(0) = \displaystyle \int_{0}^{a} \dfrac{\pi}{2} \bigg(\dfrac{1}{x} - \dfrac{1}{x\sqrt{x+1}}\bigg) {\mathrm dx}

= πln(x+1+1)0a=πln(1+a+12)\pi \ln(\sqrt{x+1} +1)\bigg|_{0}^{a} = \pi\ln\bigg(\dfrac{1+\sqrt{a+1}}{2}\bigg) (Note that I(0)=0I(0) = 0)

Put a=1/3a=1/3 , and 1/3-1/3, then substitute back in original integral to get:

I=2πln(1+4/31+2/3)=πln(7+435+26)I = 2 \pi \ln \bigg(\dfrac{1+\sqrt{4/3}}{1+\sqrt{2/3}} \bigg) = \boxed{\pi \ln \bigg(\dfrac{7 + 4\sqrt{3}}{5+2\sqrt{6}}\bigg)}

jatin yadav - 6 years, 6 months ago

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Nice solution Jatin, +1! But you're a bit late about 10 min than Ronak, so the right to propose a new problem goes to Ronak.

Anastasiya Romanova - 6 years, 6 months ago

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@Anastasiya Romanova I think maybe we should change the format of the contest. Maybe each problem should start in a new comment and the solutions must be replies to that comment. It would keep the contest more organised and understandable. The present format is just messing things up with the problem and its solution in separate comments.

Sudeep Salgia - 6 years, 6 months ago

Solution of Problem 15 :

We have 01(11x+1lnx)dx=01lnx+1x(1x)lnxdx \int_0^1 \left(\frac{1}{1-x} + \frac{1}{\ln x} \right)\,dx=\int_0^1 \frac{\ln x+1-x}{(1-x)\ln x}\,dx

Proposition :

I(s)=01slnx+1xs(1x)lnxdx=lnΓ(s+1)+γs I(s)=\int_0^1 \frac{s\ln x+1-x^s}{(1-x)\ln x}\,dx=\ln\Gamma(s+1) +\gamma s

Proof :

Let

I(s)=01slnx+1xs(1x)lnxdx I(s)=\int_0^1 \frac{s\ln x+1-x^s}{(1-x)\ln x}\,dx

then

I(s)=011xs1xdxI(s)=01xslnx1xdx=01n=0xn+slnxdx=n=0s01xn+sdx=n=0s[1n+s+1]=n=01(n+s+1)2=ψ1(s+1)I(s)=ψ1(s+1)dsI(s)=s[ψ(s+1)]dsI(s)=ψ(s+1)+C\begin{aligned} I'(s)&=\int_0^1 \frac{1-x^s}{1-x}\,dx\\ I''(s)&=-\int_0^1 \frac{x^s\ln x}{1-x} \,dx\\ &=-\int_0^1\sum_{n=0}^\infty x^{n+s}\ln x\,dx\\ &=-\sum_{n=0}^\infty\partial_s\int_0^1 x^{n+s}\,dx\\ &=-\sum_{n=0}^\infty\partial_s\left[\frac{1}{n+s+1}\right]\\ &=\sum_{n=0}^\infty\frac{1}{(n+s+1)^2}\\ &=\psi_1(s+1)\\ I'(s)&=\int\psi_1(s+1)\,ds\\ I'(s)&=\int\frac{\partial}{\partial s}\bigg[\psi(s+1)\bigg]\,ds\\ I'(s)&=\psi(s+1)+C\\ \end{aligned} For s=0s=0, we have I(0)=0I'(0)=0. Implying C=ψ(1)=γC=-\psi(1)=\gamma, then I(s)=ψ(s+1)+γI(s)=ψ(s+1)ds+γs+C=s[lnΓ(s+1)]ds+γs+C=lnΓ(s+1)+γs+C\begin{aligned} I'(s)&=\psi(s+1)+\gamma\\ I(s)&=\int \psi(s+1)\,ds +\gamma s+C\\ &=\int \frac{\partial}{\partial s}\bigg[\ln\Gamma(s+1)\bigg]\,ds +\gamma s+C\\ &=\ln\Gamma(s+1) +\gamma s+C\\ \end{aligned}

where ψ(z)\psi(z) is the digamma function. For s=0s=0, we have I(0)=0I(0)=0. Implying C=0C=0, then

I(s)=01slnx+1xs(1x)lnxdx=lnΓ(s+1)+γs I(s)=\int_0^1 \frac{s\ln x+1-x^s}{(1-x)\ln x}\,dx=\ln\Gamma(s+1) +\gamma s\qquad\qquad\square


For s=1s=1, we have

I(1)=01(11x+1lnx)dx=01lnx+1x(1x)lnxdx=γ I(1)=\int_0^1 \left(\frac{1}{1-x} + \frac{1}{\ln x} \right)\,dx=\int_0^1 \frac{\ln x+1-x}{(1-x)\ln x}\,dx=\gamma

where γ\gamma is The Euler–Mascheroni constant.

Problem 16 :

Prove

0π2cosn1xcosax dx=π2nn B(n+a+12,na+12)\int_0^{\Large\frac\pi2}\cos^{n-1}x\cos ax\ dx=\frac{\pi}{2^n n\ \operatorname{B}\left(\frac{n+a+1}{2},\frac{n-a+1}{2}\right)}

where B(x,y)\operatorname{B}\left(x,y\right) is the beta function.


PS : POST YOUR SOLUTION BELOW EACH PROBLEM THREAD AND POST YOUR PROPOSED PROBLEM AS A NEW THREAD. PUT THEM IN SEPARATED THREAD. SO THAT THE POSTS LOOK MORE ORGANIZED. THANKS.

Anastasiya Romanova - 6 years, 6 months ago

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Can you please post a all together new note of the integration contest it's getting messy and messier, Please move this contest to a new note @Anastasiya Romanova

Ronak Agarwal - 6 years, 6 months ago

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OK

The Brilliant Integration Contest - Season 1 (Part 1) moves to The Brilliant Integration Contest - Season 1 (Part 2). Enjoy it!! (>‿◠)✌

Anastasiya Romanova - 6 years, 6 months ago

Solution 9: First, consider the following Lemma.

LEMMA : 01lnx1x2dx=π28\int_{0}^{1} \frac{\ln{x}}{1-x^2} dx=-\frac{\pi^2}{8}

Proof: Since, 11x2=n=0x2n\frac{1}{1-x^2}=\sum\limits_{n=0}^{\infty} {x^{2n}}, we have,

01lnx1x2=n=001x2nlnxdx \int_0^1 \frac{\ln{x}}{1-x^2}=\sum\limits_{n=0}^{\infty} \int_0^1 x^{2n}{\ln{x}} \: dx

=n=01(2n+1)2=-\sum\limits_{n=0}^{\infty} {\frac{1}{(2n+1)^2}} (Using Integration By Parts)

=n=01(2n)2n=01(n)2=\sum\limits_{n=0}^{\infty} {\frac{1}{(2n)^2}} - \sum\limits_{n=0}^{\infty} {\frac{1}{(n)^2}}

=34ζ(2)=π28=\frac{-3}{4}\zeta(2)= -\frac{\pi^2}{8}

Now,

I=01ln(1+x1x)dxx1x2\text{I}= \int_{0}^{1} \ln({\frac{1+x}{1-x}}) \frac{dx}{x\sqrt{1-x^2}}

Substituting x=1y21+y2x=\frac{1-y^2}{1+y^2}, we have,

I=4×01lny1y2dx\text{I}=-4 \times \int_{0}^{1} \frac{\ln{y}}{1-y^2} dx

=π22=\frac{\pi^2}{2} (Using the Lemma)

Ishan Singh - 6 years, 6 months ago

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Nice answer, +1! But I'm really sorry, the one who deserves to propose the next problem is Ronak since according to the timeline his answer comes first than the other answers. Maybe, you can try to answer his proposed problem. ¨\ddot\smile

Anastasiya Romanova - 6 years, 6 months ago

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@Anastasiya Romanova I think maybe we should change the format of the contest. Maybe each problem should start in a new comment and the solutions must be replies to that comment. It would keep the contest more organised and understandable. The present format is just messing things up with the problem and its solution in separate comments.

Sudeep Salgia - 6 years, 6 months ago

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@Sudeep Salgia Ya, I agree with you. I have edited the rule accordingly. Well, this is the first contest here and I hope you understand with that.

Anastasiya Romanova - 6 years, 6 months ago

Solution of Problem 5 :

I will prove the general form of

0xm11+xndx=πnsinmπn\int_0^\infty\frac{x^{m-1}}{1+x^n}\,dx=\frac{\pi}{n\sin\frac{m\pi}{n}}

It can easily be proven by taking substitution y=11+xn\displaystyle y=\frac{1}{1+x^n} and the integral becomes Beta function

1n01y1mn1 (1y)mn1dy=Γ(1mn)Γ(mn)n=πnsinmπn\frac{1}{n}\int_0^1 y^{\large 1-\frac{m}{n}-1}\ (1-y)^{\large \frac{m}{n}-1}\,dy=\frac{\Gamma\left(1-\frac{m}{n}\right)\Gamma\left(\frac{m}{n}\right)}{n}=\frac{\pi}{n\sin\frac{m\pi}{n}}

where the last part comes from Euler's reflection formula for the gamma function. Hence, by setting m=1m=1, we have

011+xndx=πnsin(πn)=πncsc(πn)\int_0^\infty\frac{1}{1+x^n}\,dx=\frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}=\frac{\pi}{n}\csc\left(\frac{\pi}{n}\right)

Problem 6 :

Show that

0π/2sin2xsinx+cosxdx=ln(3+22)22\int_0^{\pi/2} \frac{\sin^2 x}{\sin x + \cos x}\,dx=\frac{\ln\left(3+2\sqrt{2}\right)}{2\sqrt{2}}

Anastasiya Romanova - 6 years, 6 months ago

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I=0π/2sin2xsinx+cosxdxI = \displaystyle \int_{0}^{\pi/2} \dfrac{\sin^2 x}{\sin x + \cos x} {\mathrm dx}

I=0π/2cos2xsinx+cosxdxI = \displaystyle \int_{0}^{\pi/2} \dfrac{\cos^2 x}{\sin x + \cos x} {\mathrm dx} (As 0af(x)dx=0af(ax)dx\displaystyle \int_{0}^{a} f(x) {\mathrm dx} = \int_{0}^{a} f(a-x) {\mathrm dx} )

Hence, 2I=0π/21sinx+cosxdx2I = \displaystyle \int_{0}^{\pi/2} \dfrac{1}{\sin x + \cos x} {\mathrm dx}

Hence, I=120π/21+tan2(x/2)2tan(x/2)+1tan2(x/2)dxI = \dfrac{1}{2} \displaystyle \int_{0}^{\pi/2} \dfrac{1 + \tan^2 (x/2)}{ 2 \tan(x/2) + 1 -\tan^2(x/2)} {\mathrm dx}

Putting tanx2=t\tan \dfrac{x}{2} = t,

I=01dt2(1t)2I = \displaystyle \int_{0}^{1} \dfrac{dt}{2 - (1-t)^2}

=122ln(2+t12t+1)01\displaystyle \dfrac{1}{2 \sqrt{2}} \ln \bigg(\dfrac{\sqrt{2} + t - 1}{\sqrt{2} - t + 1} \bigg) \bigg|_{0}^{1}

= ln(3+22)22\displaystyle \dfrac{\ln(3 + 2 \sqrt{2})}{2 \sqrt{2}}

Problem 7

Prove that 0π1(10+cosx)3=7π162\displaystyle \int_{0}^{\pi} \dfrac{1}{(\sqrt{10}+ \cos x)^3} = \dfrac{7 \pi}{162}

jatin yadav - 6 years, 6 months ago

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I(a)=0π1(acos(x)),a>1 I(a) = \int_{0}^{\pi} \frac{1}{(a-\cos(x))}, a>1 Use Weierstrass substitution, its not hard to show that , this integral =πa21 =\frac{\pi}{\sqrt{a^2-1}} Now , we differentiate, both sides with respect to a a . I(a)=0π1(acos(x))2=ddaπa21 I'(a) = \int_{0}^{\pi} -\frac{1}{(a-\cos(x))^2} = \frac{d}{da} \frac{\pi}{\sqrt{a^2-1}} Again differentiating both sides, I(a)=0π2(acos(x))3=dd2aπa21 I''(a) = \int_{0}^{\pi} \frac{2}{(a-\cos(x))^3} = \frac{d}{d^2a} \frac{\pi}{\sqrt{a^2-1}} Taking, 2 to RHS gives, I(a)=0π1(acos(x))3=dd2aπ2a21 I''(a) = \int_{0}^{\pi} \frac{1}{(a-\cos(x))^3} = \frac{d}{d^2a} \frac{\pi}{2\sqrt{a^2-1}} Note that, we need I(10) I''(\sqrt{10}) . Plugging, a=10 a=\sqrt{10} , and evaluating the expression in RHS gives, 7π162 \frac{7\pi}{162}

Shivang Jindal - 6 years, 6 months ago

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@Shivang Jindal -_- . Solution was already there :||||. Please, make this thread little structured. I didn't saw any solution after problem problem -7th, so wrote solution here. After posting here , i realized that, there is exactly same solution posted before.

Shivang Jindal - 6 years, 6 months ago

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@Shivang Jindal I can do nothing since the order of the threads on Brilliant.org is based on number of upvotes and downvotes. The only thing I can do is numbering the problem & the solution. If you're interesting in participating on this contest, you can try to answer the problem (PROBLEM 9) that I've just posted below.

Anastasiya Romanova - 6 years, 6 months ago

It looks like there's someone here who is interesting in knowing the closed-form of your previous problem 7 and he posted it on Math S.E.. I am wondering, does that really have a closed-form? I tried to solve it for hours but no success & you really gave me lots of trouble cracking that integral.

Anastasiya Romanova - 6 years, 6 months ago

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@Anastasiya Romanova Well ,actually I had to change the problem as I didn't know its solution. Actually, few days back, at the same website I tried answering this problem and got struck at it.

jatin yadav - 6 years, 6 months ago

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@Jatin Yadav OMG! Why did you propose then here on this contest? Anyway, just ignore that user. He always ask two or three hard integrals in a single question on Math S.E. I can answer one of his question but only gets 3 upvotes. It's not worth enough. Luckily, he accepts my answer. I also answer one of his question on Integrals and Series, but of course not for free. LOL

Anastasiya Romanova - 6 years, 6 months ago

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@Anastasiya Romanova My bad! Didn't read the rules carefully. By the way , seeing a 14 year old student having so much knowledge of calculus is really impressive! I knew only basic integration properly when I was 14.

jatin yadav - 6 years, 6 months ago

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@Jatin Yadav No worries. It's so-so in my country. I learned from my late grandpa, my big brother (I hate him!), and lots of people on internet since I was 7. You might be interested in seeing this. You're very good at all math subjects & physics too. That's truly impressive!!

Anyway, I've just remembered that the same user (Samurai, the one who posts your previous problem 7 on Math S.E.) also posted one of my problem on Brilliant at Math S.E. too. What a cheater!

Anastasiya Romanova - 6 years, 6 months ago

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@Anastasiya Romanova If I'm not mistaken, you only solve / post problems in the calculus section. Why are you specifically interested in Calculus? How about other fields of Math?

Christopher Boo - 6 years, 6 months ago

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@Christopher Boo Because I wanna beat my big brother. Seriously, he always says to me that calculus is the highest level mathematics, so that's why I'm really interested in calculus specially integrals & series and almost every weekend we have a competition of solving integrals & series problems although he always beats me easily. Of course I love math, it means I also love the other fields of math but I'm just not too interested in them. Besides, my first crush on math because of calculus. I don't wanna tell the story because it's too personal. Special for you, I have answered two problems here that don't relate to Calculus to show my capability other than Calculus (It's also because I'm bored). Here they are Algebra problem: Advanced System Of Equations and Geometry problem: All Three Concurrency. Answering Algebra problem makes me level up, so I must down my level again. I just want the other know me because I'm sorta good at Calculus, not the other subjects. Knowing Calculus in such young age also gives me lots of advantages on the other forums where I join in. Okay, I think it's enough. I hope I am not making any immodest statements or showing narcissism on my part. ¨\ddot\smile

Anastasiya Romanova - 6 years, 6 months ago

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@Anastasiya Romanova Tip for beating your older brother: Calculus is far from the hardest discipline of mathematics. Learn abstract algebra, or complex analysis, or dynamical systems. Then you'll be able to appreciate fields of mathematics where there is still a lot of work left to be done, even today, and you'll probably end up knowing more math than your big brother. ;)

Michael Lee - 6 years, 6 months ago

Ans:ln4a/2a^2

Shreyas Dighe - 6 years, 6 months ago
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