Question:
Mark the correct alternative in each of the following:
Evaluate $\int \frac{\sin ^{2} x}{\cos ^{4} x} d x=$
A. $\frac{1}{3} \tan ^{2} \mathrm{x}+\mathrm{C}$
B. $\frac{1}{2} \tan ^{2} \mathrm{x}+\mathrm{C}$
C. $\frac{1}{3} \tan ^{3} \mathrm{x}+\mathrm{C}$
D. none of these
Solution:
$I=\int(\tan x)^{2}(\sec x)^{2} d x$
$\Rightarrow \tan \mathrm{x}=\mathrm{t}\left[\frac{d t}{d x}=(\sec x)^{2}\right]$
$\Rightarrow \int t^{2} d t=\frac{t^{3}}{3}+c$
$\Rightarrow I=\frac{1}{3}(\tan x)^{3}+c$