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受欢迎的微积分 >

integral of sin^3(4x)sqrt(cos(4x))

解答

\int sin^{3} (4 x) cos (4 x) d x

解答

- \frac{1}{4} (\frac{2}{3} cos^{\frac{3}{2}} (4 x) - \frac{2}{7} cos^{\frac{7}{2}} (4 x)) + C

求解步骤

\int sin^{3} (4 x) cos (4 x) d x

使用换元积分法

= \int sin^{3} (u) cos (u) \frac{1}{4} d u

提出常数:

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

= \frac{1}{4} \cdot \int sin^{3} (u) cos (u) d u

使用换元积分法

= \frac{1}{4} \cdot \int - v (1 - v^{2}) d v

提出常数:

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

= \frac{1}{4} (- \int v (1 - v^{2}) d v)

乘开

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

= \frac{1}{4} (- \int v - v^{\frac{5}{2}} d v)

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

= \frac{1}{4} (- (\int v d v - \int v^{\frac{5}{2}} d v))

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

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

= \frac{1}{4} (- (\frac{2}{3} v^{\frac{3}{2}} - \frac{2}{7} v^{\frac{7}{2}}))

代回

= \frac{1}{4} (- (\frac{2}{3} cos^{\frac{3}{2}} (4 x) - \frac{2}{7} cos^{\frac{7}{2}} (4 x)))

化简

= - \frac{1}{4} (\frac{2}{3} cos^{\frac{3}{2}} (4 x) - \frac{2}{7} cos^{\frac{7}{2}} (4 x))

解答补常数

= - \frac{1}{4} (\frac{2}{3} cos^{\frac{3}{2}} (4 x) - \frac{2}{7} cos^{\frac{7}{2}} (4 x)) + C

作图

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