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Find the integral of (1+3 cos2x)1/2 sin (2x)

by RoRi

Evaluate the following integral:

    \[ \int \sqrt{1+3 \cos^2 x} \sin (2x) \, dx. \]

Here we make the substitution (recalling the trig identity \sin (2x) = 2 \sin x \cos x),

    \[ u = 1+3 \cos^2 x \quad \implies \quad du = -6 \cos x \sin x \, dx= -3 \sin (2x) \, dx. \]

Thus, we have

    \begin{align*} \int \sqrt{1+3 \cos^2 x} \sin (2x) \, dx &= -\frac{1}{3} \int u^{\frac{1}{2}} \, du \\ &= -\frac{2}{9} u^{\frac{3}{2}} + C \\ &= -\frac{2}{9} (1+3\cos^2 x)^{\frac{3}{2}} + C. \end{align*}

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