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受欢迎的微积分 >

integral of 3sin^5(x)cos^{-3/2}(x)

解答

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

解答

3 (- \frac{2}{5} cos^{\frac{5}{2}} (x) + \frac{4}{9} cos^{\frac{9}{2}} (x) - \frac{2}{13} cos^{\frac{13}{2}} (x)) + C

求解步骤

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

提出常数:

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

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

- \frac{3}{2} = 3 \cdot \frac{1}{2}

= 3 \cdot \int sin^{5} (x) cos^{3 \cdot \frac{1}{2}} (x) d x

乘

3 \cdot \frac{1}{2} : \frac{3}{2}

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

使用三角恒等式改写

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

使用换元积分法

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

乘开

- u^{\frac{3}{2}} (1 - u^{2})^{2} : - u^{\frac{3}{2}} + 2 u^{\frac{7}{2}} - u^{\frac{11}{2}}

= 3 \cdot \int - u^{\frac{3}{2}} + 2 u^{\frac{7}{2}} - u^{\frac{11}{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

= 3 (- \int u^{\frac{3}{2}} d u + \int 2 u^{\frac{7}{2}} d u - \int u^{\frac{11}{2}} d u)

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

\int 2 u^{\frac{7}{2}} d u = \frac{4}{9} u^{\frac{9}{2}}

\int u^{\frac{11}{2}} d u = \frac{2}{13} u^{\frac{13}{2}}

= 3 (- \frac{2}{5} u^{\frac{5}{2}} + \frac{4}{9} u^{\frac{9}{2}} - \frac{2}{13} u^{\frac{13}{2}})

u = cos (x)

代回

= 3 (- \frac{2}{5} cos^{\frac{5}{2}} (x) + \frac{4}{9} cos^{\frac{9}{2}} (x) - \frac{2}{13} cos^{\frac{13}{2}} (x))

解答补常数

= 3 (- \frac{2}{5} cos^{\frac{5}{2}} (x) + \frac{4}{9} cos^{\frac{9}{2}} (x) - \frac{2}{13} cos^{\frac{13}{2}} (x)) + C

作图

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