Question

integral sinx cos^3x/1+cos^2x.dx​

Answer 1

Answer:

We have to find the indefinite integral \int {\frac{\sin x}{\cos (3x)\cos (2x)}} \, dx∫cos(3x)cos(2x)sinxdx .

Now $$\begin{lgathered}\sin x=\sin (3x-2x)\\ \sin x=\sin (3x)\cos (2x)-\cos (3x) \sin (2x)\end{lgathered}$$

The integral becomes,

$$\begin{lgathered}\int {\frac{\sin x}{\cos (3x)\cos (2x)}} \, dx=\int {\frac{\sin (3x)\cos (2x)-\cos (3x) \sin (2x)}{\cos (3x)\cos (2x)}} \, dx\\ \int {\frac{\sin x}{\cos (3x)\cos (2x)}} \, dx=\int [{\tan (3x)-\tan(2x)] \, dx\\\end{lgathered}$$

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