integral from 0 to pi of (3-sin(x))^6

解答

\int_{0}^{π} (3 - sin (x))^{6} d x

\frac{110995 π - 292416}{80}

十进制

703.56345 \dots

求解步骤

\int_{0}^{π} (3 - sin (x))^{6} d x

乘开

(3 - sin (x))^{6} : 729 - 1458 sin (x) + 1215 sin^{2} (x) - 540 sin^{3} (x) + 135 sin^{4} (x) - 18 sin^{5} (x) + sin^{6} (x)

= \int_{0}^{π} 729 - 1458 sin (x) + 1215 sin^{2} (x) - 540 sin^{3} (x) + 135 sin^{4} (x) - 18 sin^{5} (x) + sin^{6} (x) d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{π} 729 d x - \int_{0}^{π} 1458 sin (x) d x + \int_{0}^{π} 1215 sin^{2} (x) d x - \int_{0}^{π} 540 sin^{3} (x) d x + \int_{0}^{π} 135 sin^{4} (x) d x - \int_{0}^{π} 18 sin^{5} (x) d x + \int_{0}^{π} sin^{6} (x) d x

\int_{0}^{π} 729 d x = 729 π

\int_{0}^{π} 1458 sin (x) d x = 2916

\int_{0}^{π} 1215 sin^{2} (x) d x = \frac{1215 π}{2}

\int_{0}^{π} 540 sin^{3} (x) d x = 720

\int_{0}^{π} 135 sin^{4} (x) d x = \frac{405 π}{8}

\int_{0}^{π} 18 sin^{5} (x) d x = \frac{96}{5}

\int_{0}^{π} sin^{6} (x) d x = \frac{5 π}{16}

= 729 π - 2916 + \frac{1215 π}{2} - 720 + \frac{405 π}{8} - \frac{96}{5} + \frac{5 π}{16}

化简

729 π - 2916 + \frac{1215 π}{2} - 720 + \frac{405 π}{8} - \frac{96}{5} + \frac{5 π}{16} : \frac{110995 π - 292416}{80}

= \frac{110995 π - 292416}{80}

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