Question

Evaluate:

\( \int \frac{\cos x}{\sin ^{2} x+3 \sin x+2} d x \)

Answer 1

The given integral is

\(\int \frac{cos x dx}{sin^2 x + 3 sin x + 2}\)

\(= \int \frac{dt}{t^2 + 3t + 2}\)

Let t = sin x

dt = cos x dx

\(= \int \frac{dt}{(t + 1)(t+2)}\)

\(= \int (\frac{1}{t + 1} - \frac{1}{t + 2}) dt\)

\(= log |\frac{t +1}{t+2}| + c\)

\(= log |\frac{sinx + 1}{sinx + 2}| + c\)