integral from 0 to pi/3 of xsin(12x)

解答

\int_{0}^{\frac{π}{3}} x sin (12 x) d x

- \frac{π}{36}

十进制

- 0.08726 \dots

求解步骤

\int_{0}^{\frac{π}{3}} x sin (12 x) d x

使用分布积分法

= [\frac{1}{12} (- x cos (12 x) - 12 \cdot \int - \frac{1}{12} cos (12 x) d x)]_{0}^{\frac{π}{3}}

\int - \frac{1}{12} cos (12 x) d x = - \frac{1}{144} sin (12 x)

= [\frac{1}{12} (- x cos (12 x) - 12 (- \frac{1}{144} sin (12 x)))]_{0}^{\frac{π}{3}}

化简

[\frac{1}{12} (- x cos (12 x) - 12 (- \frac{1}{144} sin (12 x)))]_{0}^{\frac{π}{3}} : [\frac{1}{12} (- x cos (12 x) + \frac{1}{12} sin (12 x))]_{0}^{\frac{π}{3}}

= [\frac{1}{12} (- x cos (12 x) + \frac{1}{12} sin (12 x))]_{0}^{\frac{π}{3}}

带入上下限计算后得:

- \frac{π}{36}

= - \frac{π}{36}

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

d er i v a t i v e f (x) = \frac{sin ^{2} ( x )}{x ^{2}}

\frac{d y}{d x} = (y - 1)^{- 3}

\int \frac{sin ( θ )}{cos ^{2} ( θ ) - cos ( θ ) - 2} d θ

d er i v a t i v e y (θ) = cos (θ^{2})