Question

Evaluate:∫cosx/((2+sinx)(3+4sinx))dx

Answer 1

We have to evaluate $\int {\frac{\cos x}{(2+\sin x)(3+4\sin x)}} \, dx$.

Make the substitution, $\sin x=t\\ \cos x dx=dt$

The integral becomes,

$\int {\frac{\cos x}{(2+\sin x)(3+4\sin x)}} \, dx =\int {\frac{dt}{(2+t)(3+4t)}} \\ \int {\frac{\cos x}{(2+\sin x)(3+4\sin x)}} \, dx =\frac{1}{5}\int [\frac{4}{3+4t}-\frac{1}{2+t}]dt\\ \int {\frac{\cos x}{(2+\sin x)(3+4\sin x)}} \, dx =\frac{1}{5} (\ln |3+4t|-\ln |2+t|)+C\\ \int {\frac{\cos x}{(2+\sin x)(3+4\sin x)}} \, dx =\frac{1}{5} \ln |\frac{3+4t}{2+t}|+C$

Back substituting,

$\int {\frac{\cos x}{(2+\sin x)(3+4\sin x)}} \, dx =\frac{1}{5} \ln |\frac{3+4\sin x}{2+\sin x}|+C$