Question

Evaluate: $\int \frac{\cos x}{3{\sin }^{2}x-4\sin x+1}dx$.

Answer 1

$\begin{array}{c}let\sin =t\\ \cos dx=dt\\ dx=\frac{dt}{\begin{array}{c}\cos x\\ \int \frac{\cos x}{3^2-4t+1}\frac{dt}{\cos x}\end{array}}\\ \frac{1}{3}\int \frac{1}{(t^2-\frac{4}{3}t+\frac{1}{3})}dt\\ \frac{1}{3}\int \frac{1}{\{t^2-2+\frac{2}{3}+\frac{4}{9}(-\frac{4}{9}+\frac{1}{3}\}}\\ \frac{1}{3}\int \frac{1}{{(t-\frac{2}{3})}^2-{\left(\frac{1}{3}\right)}^2}\\ \frac{1}{3}\frac{1}{2\times \frac{1}{3}}\log \left|\frac{(t-\frac{2}{3}+\frac{1}{3})}{t-\frac{2}{3}-\frac{1}{3}}\right|\\ \frac{1}{2}\log \left|\frac{t-\frac{1}{3}}{t-1}\right|\frac{1}{2}\log \left|\frac{\sin x-\frac{1}{3}}{\sin x-1}\right|\end{array}$