Question

integrate the function x³sin(tan^{-1}x⁴)/(1 + x^8).dx

Answer 1

HELLO DEAR,

GIVEN FUNCTION IS $\bold{\int{\frac{x^3sin(tan^{-1}x^4)}{(1 + x^8)}}\,dx}$

let tan^{-1}x⁴ = t $\bold{\Rightarrow dt/dx = x^3/(1 + x^8)}$
$\bold{dx = (1 + x^8)/x^3\,dt}$

now, $\bold{\int{\frac{x^3sin(tan^{-1}x^4)}{(1 + x^8)}*\frac{(1 + x^8)}\,dt}{x^3}}$

$\bold{\int{tan^{-1}x^4}\,dt}$

$\bold{\int{t}\,dt}$

$\bold{t^2/2 + c}$

put the value of t in above function

$\bold{(tan^{-1}x^4)/2 + c}$

where, c is arbitrary constant.

I HOPE ITS HELP YOU DEAR,
THANKS