Home » Blog » Prove an integral identity for even integer powers of cosine

Prove an integral identity for even integer powers of cosine

by RoRi

For any n \in \mathbb{Z}_{>0} prove that

    \[ \int_0^1 (1-x^2)^{n-\frac{1}{2}} \, dx = \int_0^{\frac{\pi}{2}} \cos^{2n} u \, du. \]

Proof. We make the substitution

    \[ x = \sin u \qquad dx = \cos u \, du. \]

Then, evaluating the integral

    \begin{align*} \int_0^1 (1-x^2)^{n - \frac{1}{2}} &= \int_0^{\frac{\pi}{2}} (1 - \sin^2 u)^{n - \frac{1}{2}} \cos u \, du \\ &= \int_0^{\frac{\pi}{2}} (\cos^2 u)^{n - \frac{1}{2}} \cos u \, du \\ &= \int_0^{\frac{\pi}{2}} (\cos^{2n-1} u) \cos u \, du \\ &= \int_0^{\frac{\pi}{2}} \cos^{2n} u \, du. \qquad \blacksquare \end{align*}

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):