Make a substitution to express the integrand as a rational function and then evaluate the integral

Question

Make a substitution to express the integrand as a rational function and then evaluate the integral

∫ 3cos(𝑥) / sin^2(𝑥)+sin(𝑥) dx

Note: Use an upper-case "C" for the constant of integration

Answer 1

3∫[cosx / (sin²x + sinx)]dx

Let u = sinx. Then du = cosxdx

So, the integral can be expressed as 3∫ 1 / (u(u+1)) du

Using partial fraction decomposition, 1 / (u(u+1)) = A/u + B/(u+1)

A(u+1) + Bu = 1 So, (A+B)u + A = 0u + 1. Therefore, A = 1 and B = -1.

Our integral now becomes: 3∫[1/u - 1/(u+1)]du = 3 ln l u / (u+1) l + C = 3ln l sinx / (sinx + 1) l + C