Integrals Involving Roots — Question 2

Question 2 – Solution

Step 1: Substitute $u=x+2$. Then $$x=u-2,\qquad du=dx.$$ Step 2: Rewrite and expand the integrand. $$\begin{align*} \int x\sqrt{x+2}\,dx &=\int (u-2)u^{1/2}\,du\\ &=\int\left(u^{3/2}-2u^{1/2}\right)du. \end{align*}$$ Step 3: Integrate each power. $$\begin{align*} I&=\frac{u^{5/2}}{5/2}-2\frac{u^{3/2}}{3/2}+C\\ &=\frac25u^{5/2}-\frac43u^{3/2}+C. \end{align*}$$ Step 4: Substitute $u=x+2$. $$\boxed{\frac25(x+2)^{5/2}-\frac43(x+2)^{3/2}+C}$$