Substitution Rule for Indefinite Integrals — Question 3

Question 3 - Solution

Use substitution. Let $$u = 1+x^4.$$ Then $$du = 4x^3\,dx \quad\Rightarrow\quad x^3\,dx = \frac{1}{4}\,du.$$

Substitute into the integral: $$\int \frac{x^3}{\sqrt{1+x^4}}\,dx = \frac{1}{4}\int u^{-1/2}\,du.$$

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

Substitute back: $$\boxed{ \frac{1}{2}\sqrt{1+x^4}+C }$$