integral from 0 to pi of x(7+sin(x))

解答

\int_{0}^{π} x (7 + sin (x)) d x

\frac{7 π ^{2} + 2 π}{2}

十进制

37.68520 \dots

求解步骤

\int_{0}^{π} x (7 + sin (x)) d x

使用分布积分法

= [x (7 x - cos (x)) - \int 7 x - cos (x) d x]_{0}^{π}

\int 7 x - cos (x) d x = \frac{7 x ^{2}}{2} - sin (x)

= [x (7 x - cos (x)) - (\frac{7 x ^{2}}{2} - sin (x))]_{0}^{π}

化简

[x (7 x - cos (x)) - (\frac{7 x ^{2}}{2} - sin (x))]_{0}^{π} : [\frac{1}{2} (7 x^{2} - 2 x cos (x) + 2 sin (x))]_{0}^{π}

= [\frac{1}{2} (7 x^{2} - 2 x cos (x) + 2 sin (x))]_{0}^{π}

带入上下限计算后得:

\frac{7 π ^{2} + 2 π}{2}

= \frac{7 π ^{2} + 2 π}{2}

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