Home » Blog » Evaluate the integral using substitution Evaluate the integral using substitution by RoRi October 16, 2015 Use the method of substitution to evaluate the following integral: Let Then, Related
![\[ \int x^2 \sqrt{x+1} \, dx. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-572a7b49752bd5749a681000ed81cd68_l3.png "Rendered by QuickLaTeX.com")
![\[ u = x + 1 \qquad \implies \qquad du = dx. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-33a544cb342961da9924c496c70c2088_l3.png "Rendered by QuickLaTeX.com")
![\begin{align*} \int x^2 \sqrt{x+1} \, dx &= \int (u-1)^2 \sqrt{u} \, du \\ &= \int \left(u^{\frac{5}{2}} - 2u^{\frac{3}{2}} + u^{\frac{1}{2}} \right) \, du \\[9pt] &= \frac{2}{7} u^{\frac{7}{2}} - \frac{4}{5} u^{\frac{5}{2}} + \frac{2}{3} u^{\frac{3}{2}} + C \\[9pt] &= \frac{2}{7} (x+1)^{\frac{7}{2}} - \frac{4}{5} (x+1)^{\frac{5}{2}} + \frac{2}{3} (x+1)^{\frac{3}{2}} + C. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-e188efb0b38bf6ac5b87e05e75f3492c_l3.png "Rendered by QuickLaTeX.com")
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