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受欢迎的微积分 >

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

解答

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

解答

- \frac{1}{12} cos^{4} (3 x) + \frac{1}{6} cos^{6} (3 x) - \frac{1}{8} cos^{8} (3 x) + \frac{1}{30} cos^{10} (3 x) + C

求解步骤

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

使用换元积分法

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

使用三角恒等式改写

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

使用换元积分法

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

代回

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

化简

\frac{1}{3} (- \frac{cos ^{4} ( 3 x )}{4} + \frac{cos ^{6} ( 3 x )}{2} - \frac{3 cos ^{8} ( 3 x )}{8} + \frac{cos ^{10} ( 3 x )}{10}) : - \frac{1}{12} cos^{4} (3 x) + \frac{1}{6} cos^{6} (3 x) - \frac{1}{8} cos^{8} (3 x) + \frac{1}{30} cos^{10} (3 x)

= - \frac{1}{12} cos^{4} (3 x) + \frac{1}{6} cos^{6} (3 x) - \frac{1}{8} cos^{8} (3 x) + \frac{1}{30} cos^{10} (3 x)

解答补常数

= - \frac{1}{12} cos^{4} (3 x) + \frac{1}{6} cos^{6} (3 x) - \frac{1}{8} cos^{8} (3 x) + \frac{1}{30} cos^{10} (3 x) + C

作图

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