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受欢迎的微积分 >

integral of cos^5(3y)

解答

\int cos^{5} (3 y) d y

解答

\frac{1}{3} (sin (3 y) - \frac{2 sin ^{3} ( 3 y )}{3} + \frac{sin ^{5} ( 3 y )}{5}) + C

求解步骤

\int cos^{5} (3 y) d y

使用换元积分法

= \int cos^{5} (u) \frac{1}{3} d u

提出常数:

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

= \frac{1}{3} \cdot \int cos^{5} (u) d u

使用三角恒等式改写

= \frac{1}{3} \cdot \int (1 - sin^{2} (u))^{2} cos (u) d u

使用换元积分法

= \frac{1}{3} \cdot \int (1 - v^{2})^{2} d v

乘开

(1 - v^{2})^{2} : 1 - 2 v^{2} + v^{4}

= \frac{1}{3} \cdot \int 1 - 2 v^{2} + v^{4} d v

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{3} (\int 1 d v - \int 2 v^{2} d v + \int v^{4} d v)

\int 1 d v = v

\int 2 v^{2} d v = \frac{2 v ^{3}}{3}

\int v^{4} d v = \frac{v ^{5}}{5}

= \frac{1}{3} (v - \frac{2 v ^{3}}{3} + \frac{v ^{5}}{5})

代回

= \frac{1}{3} (sin (3 y) - \frac{2 sin ^{3} ( 3 y )}{3} + \frac{sin ^{5} ( 3 y )}{5})

解答补常数

= \frac{1}{3} (sin (3 y) - \frac{2 sin ^{3} ( 3 y )}{3} + \frac{sin ^{5} ( 3 y )}{5}) + C

作图

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