Substitution Rule for Indefinite Integrals — Question 2

Question 2

Evaluate the integral $$\int \frac{\ln(x^2+1)}{x^2+1}\,x\,dx.$$

Question 2 - Solution

Use substitution. Let $$u=\ln(x^2+1).$$ Then $$du=\frac{2x}{x^2+1}\,dx \quad\Rightarrow\quad \frac{x}{x^2+1}\,dx=\frac{1}{2}\,du.$$

Substitute: $$\int \frac{\ln(x^2+1)}{x^2+1}\,x\,dx = \frac{1}{2}\int u\,du.$$

Integrate: $$\frac{1}{2}\int u\,du=\frac{1}{4}u^2.$$

Substitute back: $$\boxed{ \frac{1}{4}\bigl(\ln(x^2+1)\bigr)^2 + C }$$