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

解答

\int_{0}^{2 π} cos^{3} (t) d t

0

求解步骤

\int_{0}^{2 π} cos^{3} (t) d t

化简

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

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

使用三角恒等式改写

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

使用分布积分法

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

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

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

化简

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

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

带入上下限计算后得:

0

= 0

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