\(\int \frac{\sin^{2} x}{\cos^{6}x} dx\) is
Answer & explanation
Correct answer: option 4
\(\begin{aligned}\int \frac{\sin^{2} x}{\cos^{6}x} dx&=\int \tan^{2} x \sec^{2} x\sec^{2} x dx\\ &=\int \tan^{2} x (1+\tan^{2} x)\sec^{2} x dx\\ &=\int t^{2}(1+t^{2})dt, \text{ where }\tan x=t, \sec^{2} x dx=dt\\ &=\frac{t^{5}}{5}+\frac{t^{3}}{3}+c\\ &=\frac{\tan^{5} x}{5}+\frac{\tan^{3}x}{3}+c\end{aligned}\)