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

解答

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

\frac{5}{4} sin^{4} (\frac{x}{5}) + C

求解步骤

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

使用换元积分法

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

提出常数:

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

= \frac{5}{3} \cdot \int 3 u d u

使用幂法则

= \frac{5}{3} \cdot \frac{3}{4} u^{\frac{4}{3}}

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

代回

= \frac{5}{3} \cdot \frac{3}{4} (sin^{3} (\frac{x}{5}))^{\frac{4}{3}}

化简

\frac{5}{3} \cdot \frac{3}{4} (sin^{3} (\frac{x}{5}))^{\frac{4}{3}} : \frac{5}{4} sin^{4} (\frac{x}{5})

= \frac{5}{4} sin^{4} (\frac{x}{5})

解答补常数

= \frac{5}{4} sin^{4} (\frac{x}{5}) + C

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