integral from 0 to pi of (x^2+3x)cos(x)

解答

\int_{0}^{π} (x^{2} + 3 x) cos (x) d x

- 2 π - 6

十进制

- 12.28318 \dots

求解步骤

\int_{0}^{π} (x^{2} + 3 x) cos (x) d x

使用分布积分法

= [(x^{2} + 3 x) sin (x) - \int (2 x + 3) sin (x) d x]_{0}^{π}

\int (2 x + 3) sin (x) d x = - cos (x) (2 x + 3) + 2 sin (x)

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

乘开

(x^{2} + 3 x) sin (x) - (- cos (x) (2 x + 3) + 2 sin (x)) : x^{2} sin (x) + 3 x sin (x) + 2 x cos (x) + 3 cos (x) - 2 sin (x)

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

带入上下限计算后得:

- 2 π - 6

= - 2 π - 6

\int_{s}^{s} 1 d x

\int_{0^{+}}^{1} \frac{1}{x ^{2}} d x

\int_{\infty}^{0} e^{2 x} d x

\int_{4}^{9} x x^{2} - 4 d x

\int_{0}^{l n (3)} e^{2 x} - e^{x} d x