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受欢迎的微积分 >

integral of sin^3(x)*sqrt(cos(x))

解答

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

解答

- \frac{2}{3} cos^{\frac{3}{2}} (x) + \frac{2}{7} cos^{\frac{7}{2}} (x) + C

求解步骤

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

使用换元积分法

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

提出常数:

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

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

乘开

u (1 - u^{2}) : u - u^{\frac{5}{2}}

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

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

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

= - (\frac{2}{3} u^{\frac{3}{2}} - \frac{2}{7} u^{\frac{7}{2}})

u = cos (x)

代回

= - (\frac{2}{3} cos^{\frac{3}{2}} (x) - \frac{2}{7} cos^{\frac{7}{2}} (x))

化简

= - \frac{2}{3} cos^{\frac{3}{2}} (x) + \frac{2}{7} cos^{\frac{7}{2}} (x)

解答补常数

= - \frac{2}{3} cos^{\frac{3}{2}} (x) + \frac{2}{7} cos^{\frac{7}{2}} (x) + C

作图

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