Substitution Rule for Indefinite Integrals — Question 8

Question 8

Evaluate the integral $$\int \frac{3x^2}{1+x^3}\,\sqrt{\ln(1+x^3)}\,dx.$$

Question 8 - Solution

Use substitution. Let $$u=\ln(1+x^3).$$ Then $$du=\frac{3x^2}{1+x^3}\,dx.$$

Substitute into the integral: $$\int \frac{3x^2}{1+x^3}\sqrt{\ln(1+x^3)}\,dx = \int \sqrt{u}\,du.$$

Integrate: $$\int u^{1/2}\,du = \frac{2}{3}u^{3/2}.$$

Substitute back: $$\boxed{ \frac{2}{3}\bigl(\ln(1+x^3)\bigr)^{3/2} + C }$$