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受欢迎的微积分 >

integral of 2x^2cos(5x)

解答

\int 2 x^{2} cos (5 x) d x

解答

\frac{2}{125} (25 x^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x))) + C

求解步骤

\int 2 x^{2} cos (5 x) d x

提出常数:

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

= 2 \cdot \int x^{2} cos (5 x) d x

使用换元积分法

= 2 \cdot \int \frac{u ^{2} cos ( u )}{125} d u

提出常数:

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

= 2 \cdot \frac{1}{125} \cdot \int u^{2} cos (u) d u

使用分布积分法

= 2 \cdot \frac{1}{125} (u^{2} sin (u) - \int 2 u sin (u) d u)

\int 2 u sin (u) d u = 2 (- u cos (u) + sin (u))

= 2 \cdot \frac{1}{125} (u^{2} sin (u) - 2 (- u cos (u) + sin (u)))

u = 5 x

代回

= 2 \cdot \frac{1}{125} ((5 x)^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x)))

化简

2 \cdot \frac{1}{125} ((5 x)^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x))) : \frac{2}{125} (25 x^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x)))

= \frac{2}{125} (25 x^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x)))

解答补常数

= \frac{2}{125} (25 x^{2} sin (5 x) - 2 (- 5 x cos (5 x) + sin (5 x))) + C

作图

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