微积分(A)随缘一题[27]

求：\(\int \sin^{n}dx\)

\[\begin{aligned} I_n =&\int \sin^ndx \\ =&-\int \sin^{n-1}d\cos x \\ =&-\sin^{n-1}x\cos x+\int \cos x(n-1)\sin^{n-2}\cos xdx \\ =&-\sin^{n-1}x\cos x+(n-1)\int\sin^{n-2}(1-\sin^2 x)dx \\ =&-\sin^{n-1}x\cos x+(n-1)I_{n-2}-(n-1)I_{n} \\ \end{aligned} \]

\[I_n=\frac{n-1}{n}I_{n-2}-\frac{1}{n}\sin^{n-1}x\cos x+C \]