Question

Integrate : $\int \frac{x}{x^3+x^2+x+1}dx$

Answer 1

$\int \frac{x}{x^3+x^2+x+1}dx$

$\int \frac{x}{x^3+x^2+x+1}dx=\int [\frac{A}{x+1}+\frac{Bx+C}{x^2+1}]dx$

$\Rightarrow x=A(x^2+1)+(Bx+C)(x+1)$

when $x=-1\to -1=A\left(\right(-1{)}^2+1)\Rightarrow A=-\frac{1}{2}$

when equating constants $0=A+C\to C=-A\Rightarrow C=\frac{1}{2}$

when equating $x^2$ terms $0=A+B\to B=-A\Rightarrow B=\frac{1}{2}$

$\therefore \int \frac{x}{x^3+x^2+x+1}dx=\int [-\frac{1}{2(x+1)}+\frac{x}{2(x^2+1)}+\frac{1}{2(x^2+1)}]dx$

$=-\frac{1}{2}\ln (x+1)+\frac{1}{2}\left[\frac{1}{2}\ln \right(x^2+1\left)\right]+\frac{1}{2}{\tan }^{-1}x+C$