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受欢迎的微积分 >

integral of sin^7(5x)cos^3(5x)

解答

\int sin^{7} (5 x) cos^{3} (5 x) d x

解答

\frac{1}{5} (- \frac{cos ^{4} ( 5 x )}{4} + \frac{cos ^{6} ( 5 x )}{2} - \frac{3 cos ^{8} ( 5 x )}{8} + \frac{cos ^{10} ( 5 x )}{10}) + C

求解步骤

\int sin^{7} (5 x) cos^{3} (5 x) d x

使用换元积分法

= \int sin^{7} (u) cos^{3} (u) \frac{1}{5} d u

提出常数:

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

= \frac{1}{5} \cdot \int sin^{7} (u) cos^{3} (u) d u

使用三角恒等式改写

= \frac{1}{5} \cdot \int (1 - cos^{2} (u))^{3} sin (u) cos^{3} (u) d u

使用换元积分法

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

乘开

- v^{3} (1 - v^{2})^{3} : - v^{3} + 3 v^{5} - 3 v^{7} + v^{9}

= \frac{1}{5} \cdot \int - v^{3} + 3 v^{5} - 3 v^{7} + v^{9} 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}{5} (- \int v^{3} d v + \int 3 v^{5} d v - \int 3 v^{7} d v + \int v^{9} d v)

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

\int 3 v^{5} d v = \frac{v ^{6}}{2}

\int 3 v^{7} d v = \frac{3 v ^{8}}{8}

\int v^{9} d v = \frac{v ^{10}}{10}

= \frac{1}{5} (- \frac{v ^{4}}{4} + \frac{v ^{6}}{2} - \frac{3 v ^{8}}{8} + \frac{v ^{10}}{10})

代回

= \frac{1}{5} (- \frac{cos ^{4} ( 5 x )}{4} + \frac{cos ^{6} ( 5 x )}{2} - \frac{3 cos ^{8} ( 5 x )}{8} + \frac{cos ^{10} ( 5 x )}{10})

解答补常数

= \frac{1}{5} (- \frac{cos ^{4} ( 5 x )}{4} + \frac{cos ^{6} ( 5 x )}{2} - \frac{3 cos ^{8} ( 5 x )}{8} + \frac{cos ^{10} ( 5 x )}{10}) + C

作图

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