integral of sqrt(cos(3t))sin(3t)

解答

\int cos (3 t) sin (3 t) d t

- \frac{2}{9} cos^{\frac{3}{2}} (3 t) + C

求解步骤

\int cos (3 t) sin (3 t) d t

使用换元积分法

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

提出常数:

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

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

使用幂法则

= - \frac{2}{3} \cdot \frac{u ^{3}}{3}

u = cos (3 t)

代回

= - \frac{2}{3} \cdot \frac{( cos ( 3 t ) ) ^{3}}{3}

化简

- \frac{2}{3} \cdot \frac{( cos ( 3 t ) ) ^{3}}{3} : - \frac{2}{9} cos^{\frac{3}{2}} (3 t)

= - \frac{2}{9} cos^{\frac{3}{2}} (3 t)

解答补常数

= - \frac{2}{9} cos^{\frac{3}{2}} (3 t) + C

\int 8 x^{2} 6 x^{3} + 5 d x

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