integral from 0 to pi of x^2sin(x)

解答

\int_{0}^{π} x^{2} sin (x) d x

π^{2} - 4

十进制

5.86960 \dots

求解步骤

\int_{0}^{π} x^{2} sin (x) d x

使用分布积分法

= [- x^{2} cos (x) - \int - 2 x cos (x) d x]_{0}^{π}

\int - 2 x cos (x) d x = - 2 (x sin (x) + cos (x))

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

化简

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

带入上下限计算后得:

π^{2} - 4

= π^{2} - 4

\int (- \frac{2}{x ^{2}} + 3 x) d x

\int \frac{1}{3 - x ^{2} - 2 x} d x

\int tan (x - 7) \frac{1}{x - 7} d x

\int e^{a x + 7} d x

\frac{\partial}{\partial y} (- x)