More Substitution Rule — Question 1

Question 1

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

Question 1 - Solution

Step 1: Choose a substitution

Set $u=1+\ln(1+x^2)$, so $du=2x/(1+x^2)\,dx$.

$$\int\frac{x}{1+x^2}\sqrt{1+\ln(1+x^2)}\,dx =\frac12\int u^{1/2}\,du=\frac13u^{3/2}+C.$$

$$\boxed{\frac13\bigl(1+\ln(1+x^2)\bigr)^{3/2}+C.}$$

This is valid for every real $x$.