Question

Evaluate $\int \frac{\cos x+3\sin x+7}{\cos x+\sin x+1}dx$

Answer 1

$\int \frac{\cos x+3\sin x+7}{\cos x+\sin x+1}dx$

$\int \frac{\cos x+\sin x+1}{\cos x+\sin x+1}+\frac{2\sin x+6}{\cos x+\sin x+1}dx$

$=\int 1-dx+\int \frac{2\sin x+6}{\cos x+\sin x+1}dx$

$\Rightarrow 2\int \frac{2u}{(\mu +1)({\mu }^2+1)}d\mu +\int \frac{6-}{\cos x+\sin x+1}dx[\because \mu =(\tan x/2\left)\right]$

$\Rightarrow 4\int \frac{\mu +1}{2({\mu }^2+1)}dx-\int \frac{1}{2(\mu -1)}dv$

$=4\left[\frac{1}{2}\ln \right({\mu }^2+1)+{\tan }^{-1}\mu -\frac{1}{2}\ln (u+1\left)\right]$

$2.\frac{1}{2}\left[\ln \left({\sec }^2\left(\frac{x}{2}\right)+{\tan }^{-1}\left(\frac{x}{2}\right)-\ln \left(\tan \frac{x}{2}+1\right)\right)\right]$

$I_2=\int \frac{6}{\cos x+\sin x+1}dx$

$=6\int \frac{1}{\mu +1}dv[\because \mu =\tan \left(\frac{x}{2}\right)]$

$=6\ln |\tan \left(\frac{x}{2}\right)+1|$

$x+\ln {\sec }^2\left(\frac{x}{2}\right)+2{\tan }^{-1}\left(\frac{x}{2}\right)+|\tan \frac{x}{2}+1|+C$