Int(sqrt(cos x))/(sin x)dx - Weebly

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Int(sqrt(cos x))/(sin x)dx

$\begingroup$ We have to evaluate the following integral: $$\int \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx$$ I tried this: I multiplied both the numerator and denominator by $\sec x$ And substituted $\tan x = t$. But after that I got stuck. The book where this is taken from gives the following as the answer: $$\ln(1+t)\frac14\ln(1+t^4)+\frac1{2\sqrt2}\ln\frac{t^2-\sqrt2t+1}{t^2+\sqrt2t+1}-\frac12\tan^{-1}t^2+c$$ where $t=\sqrt{\cot x}$ $\endgroup$ 8 $\begingroup$ I tried to solve $\int \sin^3(x)\sqrt{\cos(x)}\,dx$ by setting it equal to $$\int \cos^{1/2}(x)\left(1-\cos^2x\right)\sin(x)\,dx $$ and then making $u=\cos(x)$ and $du=-\sin(x)$. I ended up with $$\frac{-2\cos^{3/2}(x)}{3} + \frac{\cos^2(x)}{2} + C$$ but the book's answer is $$\frac{2\cos^{7/2}(x)}{7} + \frac{2\cos^{3/2}(x)}{3} + C$$ Could you give a hint as to what I'm doing wrong? Here's my full work. $\endgroup$ 3 Something went wrong. Wait a moment and try again.

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