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Evaluate the integral using substitution

by RoRi

Use the method of substitution to evaluate the following integral:

    \[ \int t (1+t)^{\frac{1}{4}} \, dt. \]

Let

    \[ u = 1 + t \quad \implies \quad du = dt. \]

Then, we can evaluate the integral,

    \begin{align*} \int t(t+1)^{\frac{1}{4}} \, dt &= \int (u-1)u^{\frac{1}{4}} \, dt \\ &= \int u^{\frac{5}{4}} \, du - \int u^{\frac{1}{4}} \, du \\[9pt] &= \frac{4}{9} u^{\frac{9}{4}} - \frac{4}{5} u^{\frac{5}{4}} + C \\[9pt] &= \frac{4}{9} (1+t)^{\frac{9}{4}} - \frac{4}{5} (1+t)^{\frac{5}{4}} + C. \end{align*}

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