Question

Integrate the following function w.r.t.$x$ :
$\frac{x^{n-1}}{\sqrt{1+4x^n}}$

Answer 1

Let $I=\int \frac{x^{n-1}}{\sqrt{1+4x^n}}.dx$
Put $x^n=t$
$\therefore n{x}^{n-1}dx=dt$
$\therefore x^{n-1}dx=\frac{dt}{n}$
$\therefore I=\int \frac{1}{\sqrt{1+4t}}.\frac{dt}{n}$
$\frac{1}{n}\int (1+4t{)}^{-\frac{1}{2}}dt$
$\frac{1}{n}\frac{(1+4t{)}^{-\frac{1}{2}}}{\frac{1}{2}}\times \frac{1}{4}+c$
$\frac{1}{2n}.\sqrt{1+4x^n}+c$