Integration by substitution

Statement

• (a) Compute the indefinite integral $\displaystyle\int x(x^2+4)^3\,dx$.
• (b) Use it to evaluate $\displaystyle\int_0^2 x(x^2+4)^3\,dx$.

Solution

(a) Substitution $u=x^2+4$, $du=2x\,dx$: $\int x(x^2+4)^3\,dx=\frac12\int u^3\,du=\frac{u^4}{8}=\frac{(x^2+4)^4}{8}+c.$

(b) $\Big[\dfrac{(x^2+4)^4}{8}\Big]_0^2=\dfrac{8^4-4^4}{8}=\dfrac{4096-256}{8}=\dfrac{3840}{8}=480$.

$\boxed{\int_0^2 x(x^2+4)^3\,dx=480}$