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Question Number 195255 by Spillover last updated on 28/Jul/23

$$\int \frac{dx}{\cos {}^{3}x\sqrt{4\sin x\cos x}}$$

Answered by Frix last updated on 28/Jul/23

$$[\mathrm{Using}t=\sqrt{\tan x}]$$$$=\int (t^4+1)dt=...=\frac{5+{\tan }^{2}x}{5}\sqrt{\tan x}+C$$

Answered by Spillover last updated on 03/Aug/23

$$\int \frac{dx}{\cos {}^{3}x\sqrt{4\sin x\cos x}}$$$$=\int \frac{dx}{2\cos {}^{4}x\sqrt{\tan x}}=\frac{1}{2}\int \frac{dx}{\cos {}^{4}x\sqrt{\tan x}}$$$$\frac{1}{2}\int \frac{dx(1+\tan {}^{2}x)\sec {}^{2}x}{\sqrt{\tan x}}$$$$u=\sqrt{\tan x}\Rightarrow \underset{spillover}{u^2=\tan x}\Rightarrow 2udu=\sec {}^{2}x$$$$\frac{1}{2}\int \frac{u^4+1}{u}.2u$$$$\int (1+u^4)du\Rightarrow u+\frac{u^5}{5}+D$$$$\sqrt{\tan x}+\frac{{\left(\sqrt{\tan x}\right)}^5}{5}+D$$

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