Learn to Fly

The following integral \[ \mathbb{A} = \displaystyle \int_{2}^{3} \dfrac{(x+3)^2+2^2}{(x+2)^2+3^2} \partial x \] has a closed-form value of \[ a + \log \left (\dfrac{b}{c} \right ) - \dfrac{\tan^{-1} \left ( \frac{d}{e} \right )}{f} \] Evaluate \( a+b+c+d+e+f \), respecting the coprime fractions.

This problem was vaguely inspired by Anastasiya Romanova's integrals. Keep it up!

#Calculus #Integrals #ClosedForm

Note by Guilherme Dela Corte
6 years, 6 months ago

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Comments

@Anastasiya Romanova: I want to learn from you!

Guilherme Dela Corte - 6 years, 6 months ago

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Learn to fly where? I don't have any intention to fly anywhere. Haha

Don't learn from me, I'm still learning too. ⌣¨\quad\ddot\smile

Anyway, we can use general expression

A=∫qp(x+p)2+q2(x+q)2+p2 dx\mathbb{A}=\int_q^p\frac{(x+p)^2+q^2}{(x+q)^2+p^2}\,dx

Making substitution y=x−qy=x-q, we get A=∫0p−q(y+p+q)2+q2y2+p2 dy=∫0p−q[y2+2(p+q)y+(p+q)2+q2y2+p2] dy=∫0p−q[y2y2+p2+2(p+q)yy2+p2+(p+q)2+q2y2+p2] dy=∫0p−q[1−p2y2+p2+2(p+q)yy2+p2+(p+q)2+q2y2+p2] dy=∫0p−q[1+2(p+q)yy2+p2+(p+q)2+q2−p2y2+p2] dy=∫0p−q[1+2(p+q)yy2+p2⏟set u=y2+p2+2(pq+q2)y2+p2⏟set y=ptan⁡θ] dy\begin{aligned} \mathbb{A}&=\int_0^{p-q}\frac{(y+p+q)^2+q^2}{y^2+p^2}\,dy\\ &=\int_0^{p-q}\left[\frac{y^2+2(p+q)y+(p+q)^2+q^2}{y^2+p^2}\right]\,dy\\ &=\int_0^{p-q}\left[\frac{y^2}{y^2+p^2}+\frac{2(p+q)y}{y^2+p^2}+\frac{(p+q)^2+q^2}{y^2+p^2}\right]\,dy\\ &=\int_0^{p-q}\left[1-\frac{p^2}{y^2+p^2}+\frac{2(p+q)y}{y^2+p^2}+\frac{(p+q)^2+q^2}{y^2+p^2}\right]\,dy\\ &=\int_0^{p-q}\left[1+\frac{2(p+q)y}{y^2+p^2}+\frac{(p+q)^2+q^2-p^2}{y^2+p^2}\right]\,dy\\ &=\int_0^{p-q}\left[1+\underbrace{\frac{2(p+q)y}{y^2+p^2}}_{\large\color{#D61F06}{\text{set}\,u=y^2+p^2}}+\underbrace{\frac{2(pq+q^2)}{y^2+p^2}}_{\large\color{#3D99F6}{\text{set}\,y=p\tan\theta}}\right]\,dy\\ \end{aligned} The rest is trivial.

Anastasiya Romanova - 6 years, 6 months ago

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“I always wonder why birds choose to stay in the same place when they can fly anywhere on the earth, then I ask myself the same question.” ― Harun Yahya

I insist on learning from a learner!

Thanks a lot for the generic case answer!

Guilherme Dela Corte - 6 years, 6 months ago

@Anastasiya Romanova Madam can you suggest me any good book to increase my problem solving ability in calculas

sandeep Rathod - 6 years, 6 months ago
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