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受欢迎的微积分 >

integral of 5sin^3(x)cos^2(x)

解答

\int 5 sin^{3} (x) cos^{2} (x) d x

解答

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

求解步骤

\int 5 sin^{3} (x) cos^{2} (x) d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int sin^{3} (x) cos^{2} (x) d x

使用三角恒等式改写

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

使用换元积分法

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

乘开

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

= 5 \cdot \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

= 5 (- \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}

= 5 (- \frac{u ^{3}}{3} + \frac{u ^{5}}{5})

u = cos (x)

代回

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

化简

5 (- \frac{cos ^{3} ( x )}{3} + \frac{cos ^{5} ( x )}{5}) : - \frac{5}{3} cos^{3} (x) + cos^{5} (x)

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

解答补常数

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

作图

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