Integration Strategy — Question 3

Question 3

Use $u=x^2+1$ to evaluate: $$\int\frac{x^3}{\sqrt{x^2+1}}\,dx.$$

Question 3 – Solution

Step 1: Separate one factor of $x$. $$x^3\,dx=x^2(x\,dx).$$ Step 2: Substitute $u=x^2+1$. Then $$du=2x\,dx,\qquad x\,dx=\frac12du,\qquad x^2=u-1.$$ Step 3: Rewrite and simplify. $$\begin{align*} I&=\frac12\int\frac{u-1}{\sqrt{u}}\,du\\ &=\frac12\int\left(u^{1/2}-u^{-1/2}\right)du. \end{align*}$$ Step 4: Integrate each power. $$\begin{align*} I&=\frac12\left(\frac{u^{3/2}}{3/2}-\frac{u^{1/2}}{1/2}\right)+C\\ &=\frac13u^{3/2}-u^{1/2}+C. \end{align*}$$ Step 5: Return to $x$. $$\boxed{\frac13(x^2+1)^{3/2}-\sqrt{x^2+1}+C}$$