Question

∫cos^(2n+1)xdx from 0 to pi

Answer 1

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Explanation:

Answer 2

Assuming
`n\in\mathbb Z`:


`\displaystyle\int_0^\pi\cos^(2n+1)x\,\mathrm dx=\int_0^\pi\cos^(2n)x\cos x\,\mathrm dx`

`=\displaystyle\int_0^\pi(1-\sin^2x)^n\cos x\,\mathrm dx`

Let
`y=\sin x`, so that
`\mathrm dy=\cos x\,\mathrm dx`. The integral is then


`=\displaystyle\int_(\sin0=0)^(\sin\pi=0)(1-y^2)^n\,\mathrm dy=0`