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受欢迎的微积分 >

integral of 3sin^3(2x)cos^3(2x)

解答

\int 3 sin^{3} (2 x) cos^{3} (2 x) d x

解答

\frac{3}{2} (- \frac{1}{4} cos^{4} (2 x) + \frac{1}{6} cos^{6} (2 x)) + C

求解步骤

\int 3 sin^{3} (2 x) cos^{3} (2 x) d x

提出常数:

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

= 3 \cdot \int sin^{3} (2 x) cos^{3} (2 x) d x

使用换元积分法

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

提出常数:

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

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

使用三角恒等式改写

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

使用换元积分法

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

乘开

- v^{3} (1 - v^{2}) : - v^{3} + v^{5}

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

\int v^{3} d v = \frac{v ^{4}}{4}

\int v^{5} d v = \frac{v ^{6}}{6}

= 3 \cdot \frac{1}{2} (- \frac{v ^{4}}{4} + \frac{v ^{6}}{6})

代回

= 3 \cdot \frac{1}{2} (- \frac{cos ^{4} ( 2 x )}{4} + \frac{cos ^{6} ( 2 x )}{6})

化简

3 \cdot \frac{1}{2} (- \frac{cos ^{4} ( 2 x )}{4} + \frac{cos ^{6} ( 2 x )}{6}) : \frac{3}{2} (- \frac{1}{4} cos^{4} (2 x) + \frac{1}{6} cos^{6} (2 x))

= \frac{3}{2} (- \frac{1}{4} cos^{4} (2 x) + \frac{1}{6} cos^{6} (2 x))

解答补常数

= \frac{3}{2} (- \frac{1}{4} cos^{4} (2 x) + \frac{1}{6} cos^{6} (2 x)) + C

作图

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