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Evaluate the integral using substitution

by RoRi

Use the method of substitution to evaluate the following integral:

    \[ \int \frac{x^5 \, dx}{\sqrt{1-x^6}}. \]

Let

    \[ u = 1 - x^6 \quad \implies \quad du = -6x^5 \, dx. \]

Then, we can evaluate the integral,

    \begin{align*} \int \frac{x^5 \, dx}{\sqrt{1-x^6}} &= -\frac{1}{6} \int u^{-\frac{1}{2}} \, du \\ &= -\frac{1}{3} u^{\frac{1}{2}} + C \\ &= -\frac{1}{3} (1 - x^6)^{\frac{1}{2}} + C. \end{align*}

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