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Evaluate the integral using substitution

by RoRi

Use the method of substitution to evaluate the following integral:

    \[ \int z(z-1)^{\frac{1}{3}} \, dz. \]

Let

    \[ u = z - 1 \quad \implies \quad du = dz. \]

Then, we can evaluate the integral,

    \begin{align*} \int z(z-1)^{\frac{1}{3}} &= \int (u+1)u^{\frac{1}{3}} \, du \\ &= \int u^{\frac{4}{3}} \, du + \int u^{\frac{1}{3}} \, du \\ &= \frac{3}{7} u^{\frac{7}{3}} + \frac{3}{4} u^{\frac{4}{3}} + C\\ &= \frac{3}{7} (z-1)^{\frac{7}{3}} + \frac{3}{4} (z-1)^{\frac{4}{3}} + C. \end{align*}

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