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**e^(i*pi)** It's easier to use some identities and a u-sub: $\displaystyle \int \cos^5(x) = \int \cos^4(x) \cos(x)$ and that $\displaystyle \cos^2(x) = 1-\sin^2(x) \implies \cos^4(x) = (1-\sin^2(x))^2$

Let $\displaystyle u = \sin(x) \implies \dfrac{du}{dx} = \cos(x) \text{ and } dx = \dfrac{du}{\cos(x)}$ .

$\displaystyle \displaystyle \int \cos^5(x) = \int \cos^4(x) \cos(x) = \int (1-u^2)^2 \cos(x) \dfrac{du}{\cos(x)} = \int (1-u^2)^2 du$ which you can integrate simply enough.

This should work for any odd integer power of $\displaystyle \cos(x)$