# Chapter 8 - Techniques of Integration - 8.6 Strategies for Integration - Exercises - Page 431: 39

$$2\left( \frac{1}{5}(x+1)^{5/2} -\frac{2}{3}(x+1)^{3/2}+\sqrt{x+1}\right)+C$$

#### Work Step by Step

Given $$\int \frac{x^{2}}{\sqrt{x+1}} d x$$ Let $$u^2 =x+1\ \ \ \ \ \ \ 2udu=dx$$ Then \begin{align*} \int \frac{x^{2}}{\sqrt{x+1}} d x&=2\int \frac{(u^2-1)^{2}}{u} udu\\ &= 2\int (u^4-2u^2+1)du\\ &= 2\left( \frac{1}{5}u^5 -\frac{2}{3}u^3+u\right)+C \\ &= 2\left( \frac{1}{5}(x+1)^{5/2} -\frac{2}{3}(x+1)^{3/2}+\sqrt{x+1}\right)+C \end{align*}

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