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受欢迎的微积分 >

integral of (sin(x))^6(cos(x))^5

解答

\int (sin (x))^{6} (cos (x))^{5} d x

解答

\frac{sin ^{7} ( x )}{7} - \frac{2 sin ^{9} ( x )}{9} + \frac{sin ^{11} ( x )}{11} + C

求解步骤

\int (sin (x))^{6} (cos (x))^{5} d x

化简

= \int sin^{6} (x) cos^{5} (x) d x

使用三角恒等式改写

= \int (1 - sin^{2} (x))^{2} cos (x) sin^{6} (x) d x

使用换元积分法

= \int u^{6} (1 - u^{2})^{2} d u

乘开

u^{6} (1 - u^{2})^{2} : u^{6} - 2 u^{8} + u^{10}

= \int u^{6} - 2 u^{8} + u^{10} d u

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

= \int u^{6} d u - \int 2 u^{8} d u + \int u^{10} d u

\int u^{6} d u = \frac{u ^{7}}{7}

\int 2 u^{8} d u = \frac{2 u ^{9}}{9}

\int u^{10} d u = \frac{u ^{11}}{11}

= \frac{u ^{7}}{7} - \frac{2 u ^{9}}{9} + \frac{u ^{11}}{11}

u = sin (x)

代回

= \frac{sin ^{7} ( x )}{7} - \frac{2 sin ^{9} ( x )}{9} + \frac{sin ^{11} ( x )}{11}

解答补常数

= \frac{sin ^{7} ( x )}{7} - \frac{2 sin ^{9} ( x )}{9} + \frac{sin ^{11} ( x )}{11} + C

作图

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