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Evaluate the integral using substitution

by RoRi

Use the method of substitution to evaluate the following integral:

    \[ \int_3^8 \frac{\sin \left( \sqrt{x + 1} \right)}{\sqrt{x+1}} \, dx. \]

Let

    \[ u = \sqrt{x+1} \quad \implies \quad du = \frac{1}{2} (x+1)^{-\frac{1}{2}} \, dx. \]

For the limits of integration we have

    \begin{align*} u(3) &= 2 \\ u(8) &= 3 \end{align*}

Then, we can evaluate the integral,

    \begin{align*} \int_3^8 \frac{\sin \left( \sqrt{x+1} \right)}{\sqrt{x+1}} \, dx &= 2 \int_2^3 \sin u \, du \\ &= 2 \left( -\cos u \Bigr \rvert_2^3 \right) \\ &= 2 (\cos 2 - \cos 3). \end{align*}

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