Computing Indefinite Integrals — Question 3

Question 3 - Solution

Rewrite the integrand by separating powers of $x$: $$\frac{x^3}{(x^2+1)^2} = \frac{x(x^2+1-1)}{(x^2+1)^2} = \frac{x}{x^2+1}-\frac{x}{(x^2+1)^2}.$$

Split the integral: $$\int \frac{x}{x^2+1}\,dx -\int \frac{x}{(x^2+1)^2}\,dx.$$

For the first integral, let $u=x^2+1$, $du=2x\,dx$: $$\int \frac{x}{x^2+1}\,dx = \frac12\int \frac{du}{u} = \frac12\ln(x^2+1).$$

For the second integral, use the same substitution: $$\int \frac{x}{(x^2+1)^2}\,dx = \frac12\int u^{-2}\,du = -\frac{1}{2u} = -\frac{1}{2(x^2+1)}.$$

Combine results: $$\boxed{ \frac12\ln(x^2+1) + \frac{1}{2(x^2+1)} + C }$$