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受欢迎的微积分 >

integral of 512x^2cos(8x)

解答

\int 512 x^{2} cos (8 x) d x

解答

64 x^{2} sin (8 x) - 2 (- 8 x cos (8 x) + sin (8 x)) + C

求解步骤

\int 512 x^{2} cos (8 x) d x

提出常数:

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

= 512 \cdot \int x^{2} cos (8 x) d x

使用换元积分法

= 512 \cdot \int \frac{u ^{2} cos ( u )}{512} d u

提出常数:

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

= 512 \cdot \frac{1}{512} \cdot \int u^{2} cos (u) d u

使用分布积分法

= 512 \cdot \frac{1}{512} (u^{2} sin (u) - \int 2 u sin (u) d u)

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

= 512 \cdot \frac{1}{512} (u^{2} sin (u) - 2 (- u cos (u) + sin (u)))

u = 8 x

代回

= 512 \cdot \frac{1}{512} ((8 x)^{2} sin (8 x) - 2 (- 8 x cos (8 x) + sin (8 x)))

化简

512 \cdot \frac{1}{512} ((8 x)^{2} sin (8 x) - 2 (- 8 x cos (8 x) + sin (8 x))) : 64 x^{2} sin (8 x) - 2 (- 8 x cos (8 x) + sin (8 x))

= 64 x^{2} sin (8 x) - 2 (- 8 x cos (8 x) + sin (8 x))

解答补常数

= 64 x^{2} sin (8 x) - 2 (- 8 x cos (8 x) + sin (8 x)) + C

作图

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