Question

Evaluate: $\int \frac{{\sin }^{2}x}{{\cos }^{4}x}dx$

• A. $\frac{1}{2}{\tan }^{2}x+c$
• B. $\frac{1}{2}{\cot }^{2}x+c$
• C. $\frac{1}{3}{\cot }^{3}x+c$
• D. $\frac{1}{3}{\tan }^{3}x+c$

Answer 1

The correct option is D $\frac{1}{3}{\tan }^{3}x+c$

Let $I=\int \frac{{\sin }^{2}x}{{\cos }^{4}x}dx=\int {\tan }^{2}x{\sec }^{2}xdx$
Put $\tan x=t\Rightarrow {\sec }^{2}xdx=dt$
$I=\int t^{2}dt=\frac{t^3}{3}=\frac{{\tan }^{3}x}{3}$