Question

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

Answer 1

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

$=\int (1-\frac{x}{x^2+1})(x+1)$

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

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

$=\int x.dx-\frac{1}{2}\int \frac{2x}{x^2+1}dx+\int \frac{1}{1+x^2}dx$

$=\frac{x^2}{2}-\frac{1}{2}dx(1+x^2)+{\tan }^{-1}x+c$