Integration by substitution method

To integrate the function $\int \frac{dx}{\sin(x) \cos^3(x)}$ , we can use a substitution method. Let’s solve it step by step:

1. Let’s use the substitution:

$u = \cos(x)$

$\implies du = -\sin(x) \, dx$

2. Rewriting the integral in terms of $u$ :

$\int \frac{dx}{\sin(x) \cos^3(x)} = \int \frac{-du}{u^3}$

3. Now, integrate:

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

$= \frac{u^{-2}}{-2} + C$

$= \frac{-1}{2u^2} + C$

4. Substituting back for $x$ :

$= \frac{-1}{2\cos^2(x)} + C$

So, the integral of $\frac{dx}{\sin(x) \cos^3(x)}$ is:

$\frac{-1}{2\cos^2(x)} + C$

Answer

$\int \frac{dx}{\sin(x) \cos^3(x)} = \frac{-1}{2\cos^2(x)} + C$