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Evaluate the integral using substitution

by RoRi

Use the method of substitution to evaluate the following integral:

    \[ \int \frac{(x^2 + 1 - 2x)^{\frac{1}{5}}}{1-x} \, dx. \]

Let’s simplify the integral some,

    \begin{align*} \int \frac{(x^2 + 1 - 2x)^{\frac{1}{5}}}{1-x} \, dx &= \int \frac{(x-1)^{\frac{2}{5}}}{1-x} \, dx \\ &= -\int (x-1)^{-\frac{3}{5}} \, dx \end{align*}

Now, let

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

So, we can evaluate,

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

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