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Evaluate the integral using substitution

Use the method of substitution to evaluate the following integral:

    \[ \int x^2 \sqrt{x+1} \, dx. \]


Let

    \[ u = x + 1 \qquad \implies \qquad du = dx. \]

Then,

    \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*}

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