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受欢迎的微积分 >

integral of 5xcos(4x)

解答

\int 5 x cos (4 x) d x

解答

\frac{5}{16} (4 x sin (4 x) + cos (4 x)) + C

求解步骤

\int 5 x cos (4 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 cos (4 x) d x

使用换元积分法

= 5 \cdot \int \frac{u cos ( u )}{16} d u

提出常数:

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

= 5 \cdot \frac{1}{16} \cdot \int u cos (u) d u

使用分布积分法

= 5 \cdot \frac{1}{16} (u sin (u) - \int sin (u) d u)

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

= 5 \cdot \frac{1}{16} (u sin (u) - (- cos (u)))

u = 4 x

代回

= 5 \cdot \frac{1}{16} (4 x sin (4 x) - (- cos (4 x)))

化简

5 \cdot \frac{1}{16} (4 x sin (4 x) - (- cos (4 x))) : \frac{5}{16} (4 x sin (4 x) + cos (4 x))

= \frac{5}{16} (4 x sin (4 x) + cos (4 x))

解答补常数

= \frac{5}{16} (4 x sin (4 x) + cos (4 x)) + C

作图

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