Question

Solve
$\int \frac{x^3}{x^3+1}dx$

Answer 1

We have,

$I=\int \frac{x^3}{x^3+1}dx$

$I=\int \frac{x^3+1-1}{x^3+1}dx$

$I=-\int \frac{1}{x^3+1}dx+\int 1dx$

$Byconvertinginpartialfractionandputtingtheintegralsign,weget$

$=\int \frac{2-x}{3\left(x^2-x+1\right)}dx+\int \frac{1}{3\left(x+1\right)}dx+\int 1dx=\frac{1}{6}\left(-\ln \left|\frac{4}{3}+\frac{4x^2-4x}{3}\right|+2\sqrt{3}\arctan \left(\frac{2x-1}{\sqrt{3}}\right)\right)+\frac{1}{3}\ln \left|x+1\right|+x+C$

$Hence,solved.$