integral of sin^6(x)*cos^7(x)

解答

\int sin^{6} (x) \cdot cos^{7} (x) d x

\frac{sin ^{7} ( x )}{7} - \frac{sin ^{9} ( x )}{3} + \frac{3 sin ^{11} ( x )}{11} - \frac{sin ^{13} ( x )}{13} + C

求解步骤

\int sin^{6} (x) cos^{7} (x) d x

使用三角恒等式改写

= \int (1 - sin^{2} (x))^{3} cos (x) sin^{6} (x) d x

使用换元积分法

= \int u^{6} (1 - u^{2})^{3} d u

乘开

u^{6} (1 - u^{2})^{3} : u^{6} - 3 u^{8} + 3 u^{10} - u^{12}

= \int u^{6} - 3 u^{8} + 3 u^{10} - u^{12} 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 3 u^{8} d u + \int 3 u^{10} d u - \int u^{12} d u

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

\int 3 u^{8} d u = \frac{u ^{9}}{3}

\int 3 u^{10} d u = \frac{3 u ^{11}}{11}

\int u^{12} d u = \frac{u ^{13}}{13}

= \frac{u ^{7}}{7} - \frac{u ^{9}}{3} + \frac{3 u ^{11}}{11} - \frac{u ^{13}}{13}

u = sin (x)

代回

= \frac{sin ^{7} ( x )}{7} - \frac{sin ^{9} ( x )}{3} + \frac{3 sin ^{11} ( x )}{11} - \frac{sin ^{13} ( x )}{13}

解答补常数

= \frac{sin ^{7} ( x )}{7} - \frac{sin ^{9} ( x )}{3} + \frac{3 sin ^{11} ( x )}{11} - \frac{sin ^{13} ( x )}{13} + C

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