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Evaluate the integral using substitution

by RoRi

Use the method of substitution to evaluate the following integral:

    \[ \int x^2 (8x^3 + 27)^{\frac{2}{3}} \, dx. \]

Let

    \[ u = 8x^3 + 27 \quad \implies \quad du = 24x^2 \, dx. \]

Then, we can evaluate the integral,

    \begin{align*} \int x^2 (8x^3 + 27)^{\frac{2}{3}} \, dx &= \frac{1}{24} \int u^{\frac{2}{3}} \, du \\ &= \frac{1}{40} u^{\frac{5}{3}} + C \\ &= \frac{1}{40} (8x^3 + 27)^{\frac{5}{3}} + C. \end{align*}

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