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受欢迎的微积分 >

integral of cos^3(3y)

解答

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

解答

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

求解步骤

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

使用换元积分法

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

使用三角恒等式改写

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

使用换元积分法

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

\int 1 d v = v

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

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

代回

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

解答补常数

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

作图

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