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受欢迎的微积分 >

integral from 0 to pi of cos^3(5x)

解答

\int_{0}^{π} cos^{3} (5 x) d x

解答

0

求解步骤

\int_{0}^{π} cos^{3} (5 x) d x

使用换元积分法

= \int_{0}^{5 π} cos^{3} (u) \frac{1}{5} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{5} \cdot \int_{0}^{5 π} cos^{3} (u) d u

化简

cos^{3} (u) : cos^{2} (u) cos (u)

= \frac{1}{5} \cdot \int_{0}^{5 π} cos^{2} (u) cos (u) d u

使用三角恒等式改写

= \frac{1}{5} \cdot \int_{0}^{5 π} (1 - sin^{2} (u)) cos (u) d u

使用分布积分法

= \frac{1}{5} [- \int - sin (2 u) sin (u) d u + cos^{2} (u) sin (u)]_{0}^{5 π}

\int - sin (2 u) sin (u) d u = - \frac{2}{3} sin^{3} (u)

= \frac{1}{5} [- (- \frac{2}{3} sin^{3} (u)) + cos^{2} (u) sin (u)]_{0}^{5 π}

化简

\frac{1}{5} [- (- \frac{2}{3} sin^{3} (u)) + cos^{2} (u) sin (u)]_{0}^{5 π} : \frac{1}{5} [\frac{1}{3} (2 sin^{3} (u) + 3 cos^{2} (u) sin (u))]_{0}^{5 π}

= \frac{1}{5} [\frac{1}{3} (2 sin^{3} (u) + 3 cos^{2} (u) sin (u))]_{0}^{5 π}

带入上下限计算后得:

0

= \frac{1}{5} \cdot 0

化简

= 0

作图

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