Question

Evaluate

$\int \frac{dx}{3sinx+4cosx}$

Answer 1

${\int }^{}\frac{1}{3\sin \left(x\right)+4\cos \left(x\right)}dx$

$ApplyWeierstrassSubstitution$

$={\int }^{}\frac{1}{\frac{4\left(1-{\tan }^2\left(\frac{x}{2}\right)\right)}{{\tan }^2\left(\frac{x}{2}\right)+1}+\frac{6\tan \left(\frac{x}{2}\right)}{{\tan }^2\left(\frac{x}{2}\right)+1}}dx$

$Substituteu=\tan \frac{x}{2}$

$=-{\int }^{}\frac{1}{2u^2-3u-2}du$

$=-{\int }^{}\frac{1}{\left(u-2\right)\left(2u+1\right)}du$

$=-\left[\frac{1}{5}{\int }^{}\frac{1}{u-2}du-\frac{2}{5}{\int }^{}\frac{1}{2u+1}du\right]$

$=\frac{\ln \left(2u+1\right)}{5}-\frac{\ln \left(u-2\right)}{5}$

$Substituteu=\tan \left(\frac{x}{2}\right)$

$=\frac{\ln \left(2\tan \left(\frac{x}{2}\right)+1\right)}{5}-\frac{\ln \left(\tan \left(\frac{x}{2}\right)-2\right)}{5}$