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受欢迎的微积分 >

integral of (cos(x))^3(sin(x))^2

解答

\int (cos (x))^{3} (sin (x))^{2} d x

解答

\frac{sin ^{3} ( x )}{3} - \frac{sin ^{5} ( x )}{5} + C

求解步骤

\int (cos (x))^{3} (sin (x))^{2} d x

化简

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

使用三角恒等式改写

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

使用换元积分法

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

乘开

u^{2} (1 - u^{2}) : u^{2} - u^{4}

= \int u^{2} - u^{4} 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^{2} d u - \int u^{4} d u

\int u^{2} d u = \frac{u ^{3}}{3}

\int u^{4} d u = \frac{u ^{5}}{5}

= \frac{u ^{3}}{3} - \frac{u ^{5}}{5}

u = sin (x)

代回

= \frac{sin ^{3} ( x )}{3} - \frac{sin ^{5} ( x )}{5}

解答补常数

= \frac{sin ^{3} ( x )}{3} - \frac{sin ^{5} ( x )}{5} + C

作图

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