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

解答

\int_{0}^{2 π} (x^{2} sin (2 x)) d x

- 2 π^{2}

十进制

- 19.73920 \dots

求解步骤

\int_{0}^{2 π} x^{2} sin (2 x) d x

使用分布积分法

= [\frac{1}{2} (- x^{2} cos (2 x) - 2 \cdot \int - x cos (2 x) d x)]_{0}^{2 π}

\int - x cos (2 x) d x = - \frac{1}{4} (2 x sin (2 x) + cos (2 x))

= [\frac{1}{2} (- x^{2} cos (2 x) - 2 (- \frac{1}{4} (2 x sin (2 x) + cos (2 x))))]_{0}^{2 π}

化简

[\frac{1}{2} (- x^{2} cos (2 x) - 2 (- \frac{1}{4} (2 x sin (2 x) + cos (2 x))))]_{0}^{2 π} : [\frac{1}{4} (- 2 x^{2} cos (2 x) + 2 x sin (2 x) + cos (2 x))]_{0}^{2 π}

= [\frac{1}{4} (- 2 x^{2} cos (2 x) + 2 x sin (2 x) + cos (2 x))]_{0}^{2 π}

带入上下限计算后得:

\frac{- 8 π ^{2} + 1}{4} - \frac{1}{4}

= \frac{- 8 π ^{2} + 1}{4} - \frac{1}{4}

化简

= - 2 π^{2}

\frac{d}{d x} (cos (x) + i sin (x))

t an g e n t y = x^{3} - 3 x + 2, (2, 4)

\int_{- \infty}^{- 1} e^{- 2 x} d x

\int x arcsin (d) x d x

\int x (x^{2} + 3)^{4} d x