求不定积分 $\displaystyle\int\sin^4 x\mathrm{d}x$, $\displaystyle\int\cos^4 x\mathrm{d}x$, $\displaystyle\int\sin^6 x\mathrm{d}x$, $\displaystyle\int\cos^6 x\mathrm{d}x$.
求不定积分 $\displaystyle\int\sin^4 x\mathrm{d}x$, $\displaystyle\int\cos^4 x\mathrm{d}x$.
求不定积分 $\displaystyle\int\sin^6 x\mathrm{d}x$, $\displaystyle\int\cos^6 x\mathrm{d}x$.
一般的, 若记 $I_n=\int\sin^n x\mathrm{d}x$, $J_n=\int\cos^n x\mathrm{d}x$, 利用分部积分可证明:
\[
I_n=-\frac{1}{n}\sin^{n-1}x\cdot\cos x+\frac{n-1}{n}I_{n-2}.
\]
\[
J_n=\frac{1}{n}\sin x\cdot\cos^{n-1} x+\frac{n-1}{n}J_{n-2}.
\]