integral from 0 to pi of xsin^3(x)

解答

\int_{0}^{π} x sin^{3} (x) d x

\frac{2 π}{3}

十进制

2.09439 \dots

求解步骤

\int_{0}^{π} x sin^{3} (x) d x

使用分布积分法

= [x (- cos (x) + \frac{1}{3} cos^{3} (x)) - \int - cos (x) + \frac{1}{3} cos^{3} (x) d x]_{0}^{π}

\int - cos (x) + \frac{1}{3} cos^{3} (x) d x = - \frac{2}{3} sin (x) - \frac{1}{9} sin^{3} (x)

= [x (- cos (x) + \frac{1}{3} cos^{3} (x)) - (- \frac{2}{3} sin (x) - \frac{1}{9} sin^{3} (x))]_{0}^{π}

化简

[x (- cos (x) + \frac{1}{3} cos^{3} (x)) - (- \frac{2}{3} sin (x) - \frac{1}{9} sin^{3} (x))]_{0}^{π} : [\frac{1}{9} (- 9 x cos (x) + 3 x cos^{3} (x) + 6 sin (x) + sin^{3} (x))]_{0}^{π}

= [\frac{1}{9} (- 9 x cos (x) + 3 x cos^{3} (x) + 6 sin (x) + sin^{3} (x))]_{0}^{π}

带入上下限计算后得:

\frac{2 π}{3}

= \frac{2 π}{3}

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