Question

Integrate the following functions.
$\int \frac{x^{3}sin\left(ta{n}^{-1}{x}^4\right)}{(1+x^8)}dx$

Answer 1

$\int \frac{x^{3}sin\left(ta{n}^{-1}{x}^4\right)}{(1+x^8)}dx$
Let $ta{n}^{-1}{x}^4=t\Rightarrow \frac{1}{1+x^8}.4x^3=\frac{dt}{dx}\Rightarrow dx=\frac{(1+x^8)}{4x^3}dt$
$\therefore I=\int \frac{x^{3}sint}{(1+x^8)}.\frac{1+x^{8}dt}{4x^3}=\frac{1}{4}\int sintdt=-\frac{1}{4}cost+C=-\frac{1}{4}cos(ta{n}^{-1}{x}^4+C)$