integral of 5xsin(5x)

解答

\int 5 x sin (5 x) d x

- x cos (5 x) + \frac{1}{5} sin (5 x) + C

求解步骤

\int 5 x sin (5 x) d x

提出常数:

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

= 5 \cdot \int x sin (5 x) d x

使用分布积分法

= 5 (- \frac{1}{5} x cos (5 x) - \int - \frac{1}{5} cos (5 x) d x)

\int - \frac{1}{5} cos (5 x) d x = - \frac{1}{25} sin (5 x)

= 5 (- \frac{1}{5} x cos (5 x) - (- \frac{1}{25} sin (5 x)))

化简

5 (- \frac{1}{5} x cos (5 x) - (- \frac{1}{25} sin (5 x))) : - x cos (5 x) + \frac{1}{5} sin (5 x)

= - x cos (5 x) + \frac{1}{5} sin (5 x)

解答补常数

= - x cos (5 x) + \frac{1}{5} sin (5 x) + C

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