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Evaluate the integral using substitution

by RoRi

Use the method of substitution to evaluate the following integral:

    \[ \int_0^{\frac{\pi}{4}} \cos (2x) \sqrt{4-\sin (2x)} \, dx. \]

Let

    \[ u = 4 - \sin (2x) \quad \implies \quad du = -2\cos (2x) \, dx. \]

For the limits of integration we have

    \begin{align*} u \left( \frac{\pi}{4} \right) &= 3 \\ u (0) &= 4 \end{align*}

Then, we can evaluate the integral,

    \begin{align*} \int_0^{\frac{\pi}{4}} \cos (2x) \sqrt{4 - \sin (2x)} \, dx &= -\frac{1}{2} \int_{u(0)}^{u\left(\frac{\pi}{4}\right)} u^{\frac{1}{2}} \, du \\ &= -\frac{1}{2} \int_4^3 u^{\frac{1}{2}} \, du \\ &= -\frac{1}{3} u^{\frac{3}{2}} \Bigr \rvert_4^3 \\ &= -\frac{1}{3} \left( \sqrt{27} - 8 \right) \\ &= \frac{8}{3} - \frac{\sqrt{27}}{3} \\ &= \frac{8}{3} - \sqrt{3}. \end{align*}

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