Exercise 31, § 1

Functions of a Real Variable, III, 123

If $l_{m,n}$ is a primitive of $\sin^m x \cos^n x$ ( $m$ and $n$ are arbitrary real numbers), show that

I_{m+2,n} = -\frac{\sin^{m+1} x \cos^{n+1} x}{m + n + 2} + \frac{m + 1}{m + n + 2} I_{m,n}

is a primitive of

\sin^{m+2} x \cos^n x

m + n + 2 \neq 0

With the aid of this formula recover the formula (29) of III, p. 100 and prove the formula

\int_0^{\pi/2} \sin^{2n+1} x \, dx = \frac{2.4.6 \ldots 2n}{1.3.5 \ldots (2n+1)} \quad (n \text{ an integer } \geqslant 0).

Solution

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