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受欢迎的微积分 >

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

解答

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

解答

- \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{8} + C

求解步骤

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

使用三角恒等式改写

= \int (1 - cos^{2} (x)) sin (x) cos^{5} (x) d x

使用换元积分法

= \int - u^{5} (1 - u^{2}) d u

乘开

- u^{5} (1 - u^{2}) : - u^{5} + u^{7}

= \int - u^{5} + u^{7} 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^{5} d u + \int u^{7} d u

\int u^{5} d u = \frac{u ^{6}}{6}

\int u^{7} d u = \frac{u ^{8}}{8}

= - \frac{u ^{6}}{6} + \frac{u ^{8}}{8}

u = cos (x)

代回

= - \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{8}

解答补常数

= - \frac{cos ^{6} ( x )}{6} + \frac{cos ^{8} ( x )}{8} + C

作图

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