integral from 0 to pi of cos^3(θ)

解答

\int_{0}^{π} cos^{3} (θ) d θ

0

求解步骤

\int_{0}^{π} cos^{3} (θ) d θ

化简

cos^{3} (θ) : cos^{2} (θ) cos (θ)

= \int_{0}^{π} cos^{2} (θ) cos (θ) d θ

使用三角恒等式改写

= \int_{0}^{π} (1 - sin^{2} (θ)) cos (θ) d θ

使用分布积分法

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

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

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

化简

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

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

带入上下限计算后得:

0

= 0

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