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Evaluate the integral using substitution

by RoRi

Use the method of substitution to evaluate the following integral:

    \[ \int \sqrt{2x+1} \, dx. \]

Let

    \[ u = 2x + 1 \qquad \implies \qquad du = 2 \, dx. \]

So, using substitution

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

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