# Math Help - Four trigonometric integrals.

1. ## Four trigonometric integrals.

Hello! I'm having some troubles solving this integrals so your help would be much appreciated.

1. $\int \frac{dx}{tan^8x}$

2. $\int \frac{dx}{\sqrt[4]{sin^3xcos^5x}}$

3. $\int \frac{dx}{(sinx+cosx)^2}$

4. $\frac{\sqrt{tgx}dx}{sinxcosx}$

2. 1) $I_n=\int\frac{dx}{\tan^nx}=\int\cot^nxdx=\int\cot^ {n-2}x\cdot\cot^2xdx=\int\cot^{n-2}x\left(\frac{1}{\sin^2x}-1\right)dx=$

$=-\int\cot^{n-2}x\cdot(\cot x)'dx-I_{n-2}=-\frac{\cot^{n-1}x}{n-1}-I_{n-2}$

3) $\sin x+\cos x=\sin x+\sin\left(\frac{\pi}{2}-x\right)=2\sin\frac{\pi}{4}\cos\left(x-\frac{\pi}{4}\right)=\sqrt{2}\cos\left(x-\frac{\pi}{4}\right)$

Then, $\int\frac{dx}{2\cos^2\left(x-\frac{\pi}{4}\right)}=\frac{1}{2}\tan\left(x-\frac{\pi}{4}\right)+C$

3. 4. use the substitution

$u = \sqrt {\tan x}$