Easy integral

Calculus Level 5

If
π / 6 π / 3 d x sec x + csc x \displaystyle \int _{ \pi /6 }^{ \pi /3 }{ \frac { dx }{\sec x+\csc x } } can be represented in the form a b 2 + c 2 ln ( d + e f g ) \frac { \sqrt { a } -b }{ 2 } +\frac { \sqrt { c } }{ 2 } \ln(\sqrt { d } +\sqrt { e } -\sqrt { f } -g) then find a + b + c + d + e + f + g a+b+c+d+e+f+g

Detials and assumptions

1) a , b , c , d , e , f , g a,b,c,d,e,f,g are integers all less than 10 10 . They need not to be distinct

2) a , c , d , e , f a,c,d,e,f are not having any factor which is a perfect square.


The answer is 19.

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2 solutions

Satyajit Mohanty
Jul 10, 2015

And then the value of the expression asked turns out to be 19 \boxed{19} .

"This is how it's done"( Courtesy : Counter Strike condition zero )

Ronak Agarwal - 5 years, 11 months ago

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:D You're an avid Gamer, it seems!

Satyajit Mohanty - 5 years, 11 months ago

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Yes, I am.

Ronak Agarwal - 5 years, 11 months ago

Try this: . I tweaked this problem inspired by a problem similar to yours.

Satyajit Mohanty - 5 years, 11 months ago

Same method as mine!!!! At end I further simplified the last term as

1 2 c o s x + 1 2 s i n x 1 2 2 l n t a n ( x + π 4 2 ) -\frac {1}{2} cos x + \frac {1}{2} sin x -\frac {1}{2\sqrt {2}}ln| tan (\frac {x + \frac {\pi}{4} }{2}) |

Aniket Sanghi - 4 years, 9 months ago

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Ya even I wrote the same expression.

Spandan Senapati - 4 years, 3 months ago
Incredible Mind
Feb 11, 2015

there is long way to do the indefinite integral and a very short way.I will say shorter way.

I=(sin2x /2)/(sinx+cosx)

sinx+cosx = sqrt2 sin(pi/4 + x)

now sub pi/4+x=u

then integral becomes

cos2u/sinu forgeting constants(u should remeber to add it in the end)

I = integral 1-2sin^2u /sinu

I = integral cosecu-2sinu

I=log |cscu-cotu|+2cosu

i left out the constants.but take care of that and just put the limits.

There is a mistake @incredible mind

During substitution the numerator should be cos 2 u -\cos{2u} not cos 2 u \cos{2u} .

As sin ( 2 u π 2 ) = cos 2 u \sin{(2u - \frac{\pi}{2})} = -\cos{2u}

Imgur Imgur

Rajdeep Dhingra - 6 years, 4 months ago

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But i mentioned it already

cos2u/sinu 'forgeting constants'.......i consider constants irrelavemt in this integration...... .i am just trying to get the indefinite integral..i mentioned at the end "i left out the constants.but take care of that"..it is pain to type all that constants u know

incredible mind - 6 years, 4 months ago

Oh! I didn't know about this one. I did the long way! 2 pages long! Long way is by substituting t a n ( x 2 ) tan(\frac{x}{2}) and partial fraction!

Kartik Sharma - 6 years, 2 months ago

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Yeah, even i used the same substitution.

Samarpit Swain - 6 years, 1 month ago

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