Half Powered Integration

Calculus Level 4

∫ 1 sin ⁡ 3 x cos ⁡ 5 x d x \large\displaystyle \int \frac{1}{\sqrt{\sin^3 x \cos^5x}} \mathrm{d}x

The integral above is equals to a tan ⁡ b x + c tan ⁡ d x + constant a\tan^b x+c\tan^d x+ \text{constant} . Where a , b , c , d a,b,c,d are reals . Find the value of 3 ( a + b + c + d ) 3(a+b+c+d) .


The answer is -1.

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Sanjeet Raria by Sanjeet Raria , 27, India

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2 solutions

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Kunal Gupta
Mar 17, 2015

The integral can be written as:
I = ∫ sec ⁡ 2 x d x tan ⁡ 3 x cos ⁡ 8 x I=\displaystyle \int \dfrac{\sec^{2}x \text{d}x}{\sqrt{\tan^{3}x \cos^{8}x}} Now let tan ⁡ x = t \tan x =t , the integral becomes: I = ( 1 + t 2 ) d t t 3 = − 2 tan ⁡ 1 2 x + 2 3 tan ⁡ 3 2 x + c I=\displaystyle \dfrac{(1+t^{2}) \text{d}t}{\sqrt{t^{3}}} = \boxed{ -2\tan^{ \frac{1}{2}}x + \dfrac{2}{3} \tan^{3}{2}x +c }

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