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受欢迎的微积分 >

integral of 4xcos(5x)

解答

\int 4 x cos (5 x) d x

解答

\frac{4}{25} (5 x sin (5 x) + cos (5 x)) + C

求解步骤

\int 4 x cos (5 x) d x

提出常数:

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

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

使用换元积分法

= 4 \cdot \int \frac{u cos ( u )}{25} d u

提出常数:

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

= 4 \cdot \frac{1}{25} \cdot \int u cos (u) d u

使用分布积分法

= 4 \cdot \frac{1}{25} (u sin (u) - \int sin (u) d u)

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

= 4 \cdot \frac{1}{25} (u sin (u) - (- cos (u)))

u = 5 x

代回

= 4 \cdot \frac{1}{25} (5 x sin (5 x) - (- cos (5 x)))

化简

4 \cdot \frac{1}{25} (5 x sin (5 x) - (- cos (5 x))) : \frac{4}{25} (5 x sin (5 x) + cos (5 x))

= \frac{4}{25} (5 x sin (5 x) + cos (5 x))

解答补常数

= \frac{4}{25} (5 x sin (5 x) + cos (5 x)) + C

作图

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\int x^{2} - 7 x + 3 d x

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