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受欢迎的微积分 >

integral of 24sin^2(3x)cos^3(3x)

解答

\int 24 sin^{2} (3 x) cos^{3} (3 x) d x

解答

8 (\frac{1}{3} sin^{3} (3 x) - \frac{1}{5} sin^{5} (3 x)) + C

求解步骤

\int 24 sin^{2} (3 x) cos^{3} (3 x) d x

提出常数:

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

= 24 \cdot \int sin^{2} (3 x) cos^{3} (3 x) d x

使用换元积分法

= 24 \cdot \int sin^{2} (u) 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

= 24 \cdot \frac{1}{3} \cdot \int sin^{2} (u) cos^{3} (u) d u

使用三角恒等式改写

= 24 \cdot \frac{1}{3} \cdot \int (1 - sin^{2} (u)) cos (u) sin^{2} (u) d u

使用换元积分法

= 24 \cdot \frac{1}{3} \cdot \int v^{2} (1 - v^{2}) d v

乘开

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

= 24 \cdot \frac{1}{3} \cdot \int 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

= 24 \cdot \frac{1}{3} (\int v^{2} d v - \int v^{4} d v)

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

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

= 24 \cdot \frac{1}{3} (\frac{v ^{3}}{3} - \frac{v ^{5}}{5})

代回

= 24 \cdot \frac{1}{3} (\frac{sin ^{3} ( 3 x )}{3} - \frac{sin ^{5} ( 3 x )}{5})

化简

24 \cdot \frac{1}{3} (\frac{sin ^{3} ( 3 x )}{3} - \frac{sin ^{5} ( 3 x )}{5}) : 8 (\frac{1}{3} sin^{3} (3 x) - \frac{1}{5} sin^{5} (3 x))

= 8 (\frac{1}{3} sin^{3} (3 x) - \frac{1}{5} sin^{5} (3 x))

解答补常数

= 8 (\frac{1}{3} sin^{3} (3 x) - \frac{1}{5} sin^{5} (3 x)) + C

作图

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