Question:
Evaluate the following integrals:
$\int \sec ^{4} 2 x d x$
Solution:
Let I $=\int \sec ^{4} 2 x d x$
$\Rightarrow I=\int \sec ^{2} 2 x \sec ^{2} 2 x d x$
$\Rightarrow I=\int\left(1+\tan ^{2} 2 x\right) \sec ^{2} 2 x d x$
$\Rightarrow I=\int\left(\sec ^{2} 2 x+\tan ^{2} 2 x \sec ^{2} 2 x\right) d x$
Let $\tan 2 x=t$, then
$\Rightarrow 2 \sec ^{2} 2 x d x=d t$
$\Rightarrow I=\frac{1}{2} \int\left(1+t^{2}\right) d t$
$\Rightarrow I=\frac{1}{2} t+\frac{1}{2} \cdot \frac{1}{3} t^{3}+c$
$\Rightarrow I=\frac{1}{2} \tan 2 x+\frac{1}{6} \tan ^{3} 2 x+c$
Therefore, $\int \sec ^{4} 2 x d x=\frac{1}{2} \tan 2 x+\frac{1}{6} \tan ^{3} 2 x+c$
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.