integral of 27xsin(3x)

解答

\int 27 x sin (3 x) d x

27 (- \frac{1}{3} x cos (3 x) + \frac{1}{9} sin (3 x)) + C

求解步骤

\int 27 x sin (3 x) d x

提出常数:

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

= 27 \cdot \int x sin (3 x) d x

使用分布积分法

= 27 (- \frac{1}{3} x cos (3 x) - \int - \frac{1}{3} cos (3 x) d x)

\int - \frac{1}{3} cos (3 x) d x = - \frac{1}{9} sin (3 x)

= 27 (- \frac{1}{3} x cos (3 x) - (- \frac{1}{9} sin (3 x)))

化简

= 27 (- \frac{1}{3} x cos (3 x) + \frac{1}{9} sin (3 x))

解答补常数

= 27 (- \frac{1}{3} x cos (3 x) + \frac{1}{9} sin (3 x)) + C

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