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

解答

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

- \frac{1}{8} cos^{4} (2 x) + C

求解步骤

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

使用换元积分法

= \int - \frac{3 u}{6} d u

提出常数:

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

= - \frac{1}{6} \cdot \int 3 u d u

使用幂法则

= - \frac{1}{6} \cdot \frac{3}{4} u^{\frac{4}{3}}

u = cos^{3} (2 x)

代回

= - \frac{1}{6} \cdot \frac{3}{4} (cos^{3} (2 x))^{\frac{4}{3}}

化简

- \frac{1}{6} \cdot \frac{3}{4} (cos^{3} (2 x))^{\frac{4}{3}} : - \frac{1}{8} cos^{4} (2 x)

= - \frac{1}{8} cos^{4} (2 x)

解答补常数

= - \frac{1}{8} cos^{4} (2 x) + C

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