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Use integration by parts to evaluate an integral

by RoRi

Given that

    \[ \int_{-1}^3 (4+t^3)^{\frac{1}{2}} \, dt = 11.35, \]

evaluate the integral

    \[ \int_{-1}^3 t^3 (4+t^3)^{-\frac{1}{2}} \, dt. \]

To apply integration by parts first define

    \begin{align*} u &= t & \implies && du &= dt \\ dv &= \frac{3}{2} t^2 \cdot \frac{1}{\sqrt{4+t^3}} \, dt & \implies && v &= \sqrt{4+t^3}. \end{align*}

Then we can use the integration by parts formula to evaluate:

    \begin{align*} \int_{-1}^3 t^3 (4+t^3)^{-\frac{1}{2}} \, dt &= \frac{2}{3} \int_{-1}^3 \frac{3}{2} t^3 (4+t^3)^{-\frac{1}{2}} \, dt \\[9pt] &= \frac{2}{3} \int_{-1}^3 u \, dv \\[9pt] &= \frac{2}{3} \left( uv \Big \rvert_{-1}^3 - \int_{-1}^3 v \, du \right) \\[9pt] &= \frac{2}{3} \left( t \sqrt{4+t^3} \Big \rvert_{-1}^3 - \int_{-1}^3 \sqrt{4+t^3} \, dt \right) \\[9pt] &= \frac{2}{3} \left( 3 \sqrt{31} + \sqrt{3} - 11.35 \right). \end{align*}

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