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Question Number 125187 by bemath last updated on 08/Dec/20

$$\int \frac{dx}{\sqrt{\sin {}^{3}x}\sqrt{\cos {}^{5}x}}?$$

Answered by liberty last updated on 08/Dec/20

$$I=\int \sqrt{\frac{\left(\frac{1}{\cos {}^{3}x}\right)}{\tan {}^{3}x\cos {}^{5}x}}dx=\int \frac{dx}{\sqrt{\tan {}^{3}x}\sqrt{\cos {}^{8}x}}$$$$I=\int \frac{\sec {}^{4}x}{\sqrt{\tan {}^{3}x}}dx=\int \frac{(1+\tan {}^{2}x)\sec {}^{2}x}{\sqrt{\tan {}^{3}x}}dx$$$$letting\tan x=p\Rightarrow \sec {}^{2}xdx=dp$$$$I=\int \frac{(1+p^2)}{p^{\frac{3}{2}}}dp=\int (p^{-\frac{3}{2}}+p^{\frac{1}{2}})dp$$$$I=-2p^{-\frac{1}{2}}+\frac{2}{3}{p}^{\frac{3}{2}}+c$$$$I=-\frac{2}{\sqrt{\tan x}}+\frac{2\sqrt{\tan {}^{3}x}}{3}+c$$

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