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受欢迎的微积分 >

integral of 3cos^3(3x)

解答

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

解答

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

求解步骤

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

提出常数:

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

= 3 \cdot \int cos^{3} (3 x) d x

使用换元积分法

= 3 \cdot \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

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

使用三角恒等式改写

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

使用换元积分法

= 3 \cdot \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

= 3 \cdot \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}

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

代回

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

化简

3 \cdot \frac{1}{3} (sin (3 x) - \frac{sin ^{3} ( 3 x )}{3}) : sin (3 x) - \frac{1}{3} sin^{3} (3 x)

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

解答补常数

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

作图

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