Science:Math Exam Resources/Courses/MATH101/April 2011/Question 04 (b)/Solution 3

We apply integration by parts with $u = \frac{x^{2}}{2}$ and $d v = 2 x (1 + x^{2})^{\frac{1}{2}} d x$ . Then $d u = x d x$ and $v = \frac{2}{3} (1 + x^{2})^{\frac{3}{2}}$ . Thus

\begin{aligned} \int x^{3} \sqrt{1 + x^{2}} d x & = \int \frac{x^{2}}{2} \cdot 2 x (1 + x^{2})^{\frac{1}{2}} d x \\ = \frac{x^{2}}{2} \frac{2}{3} (1 + x^{2})^{\frac{3}{2}} - \int x \frac{2}{3} (1 + x^{2})^{\frac{3}{2}} d x \\ = \frac{x^{2}}{2} \frac{2}{3} (1 + x^{2})^{\frac{3}{2}} - \frac{1}{3} \frac{2}{5} (1 + x^{2})^{\frac{5}{2}} + C \\ = \frac{x^{2}}{3} (1 + x^{2})^{\frac{3}{2}} - \frac{2}{15} (1 + x^{2})^{\frac{5}{2}} + C \end{aligned}

Note: The fastest way to confirm that this is the same answer as we get in the solutions above, check that they both simplify to $\sqrt{1 + x^{2}} (- \frac{2}{15} + \frac{1}{15} x^{2} + \frac{1}{5} x^{4}) + C$ .