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受欢迎的微积分 >

integral of 216x^2cos(6x)

解答

\int 216 x^{2} cos (6 x) d x

解答

36 x^{2} sin (6 x) - 2 (- 6 x cos (6 x) + sin (6 x)) + C

求解步骤

\int 216 x^{2} cos (6 x) d x

提出常数:

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

= 216 \cdot \int x^{2} cos (6 x) d x

使用换元积分法

= 216 \cdot \int \frac{u ^{2} cos ( u )}{216} d u

提出常数:

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

= 216 \cdot \frac{1}{216} \cdot \int u^{2} cos (u) d u

使用分布积分法

= 216 \cdot \frac{1}{216} (u^{2} sin (u) - \int 2 u sin (u) d u)

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

= 216 \cdot \frac{1}{216} (u^{2} sin (u) - 2 (- u cos (u) + sin (u)))

u = 6 x

代回

= 216 \cdot \frac{1}{216} ((6 x)^{2} sin (6 x) - 2 (- 6 x cos (6 x) + sin (6 x)))

化简

216 \cdot \frac{1}{216} ((6 x)^{2} sin (6 x) - 2 (- 6 x cos (6 x) + sin (6 x))) : 36 x^{2} sin (6 x) - 2 (- 6 x cos (6 x) + sin (6 x))

= 36 x^{2} sin (6 x) - 2 (- 6 x cos (6 x) + sin (6 x))

解答补常数

= 36 x^{2} sin (6 x) - 2 (- 6 x cos (6 x) + sin (6 x)) + C

作图

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