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

解答

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

\frac{1}{12} sin^{4} (3 x) + C

求解步骤

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

使用换元积分法

= \int \frac{3 u}{9} d u

提出常数:

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

= \frac{1}{9} \cdot \int 3 u d u

使用幂法则

= \frac{1}{9} \cdot \frac{3}{4} u^{\frac{4}{3}}

u = sin^{3} (3 x)

代回

= \frac{1}{9} \cdot \frac{3}{4} (sin^{3} (3 x))^{\frac{4}{3}}

化简

\frac{1}{9} \cdot \frac{3}{4} (sin^{3} (3 x))^{\frac{4}{3}} : \frac{1}{12} sin^{4} (3 x)

= \frac{1}{12} sin^{4} (3 x)

解答补常数

= \frac{1}{12} sin^{4} (3 x) + C

\int (1 - 3 x + x^{3}) 3 x d x

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

\int 10000 - 2 (2 q + q^{3}) d q

\int 20 e^{4 x} d x

\int \frac{( x - 1 )}{x ^{\frac{2}{3}} + x ^{\frac{1}{3}} + 1} d x