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Evaluate the integral using substitution

by RoRi

Use the method of substitution to evaluate the following integral:

    \[ \int \frac{\sin x \, dx}{\sqrt{\cos^3 x}}. \]

Let

    \[ u = \cos x \quad \implies \quad du = -\sin x \, dx. \]

Then, we can evaluate the integral,

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

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