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受欢迎的微积分 >

integral of-3xcos(2x)

解答

\int - 3 x cos (2 x) d x

解答

- \frac{3}{4} (2 x sin (2 x) + cos (2 x)) + C

求解步骤

\int - 3 x cos (2 x) d x

提出常数:

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

= - 3 \cdot \int x cos (2 x) d x

使用换元积分法

= - 3 \cdot \int \frac{u cos ( u )}{4} d u

提出常数:

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

= - 3 \cdot \frac{1}{4} \cdot \int u cos (u) d u

使用分布积分法

= - 3 \cdot \frac{1}{4} (u sin (u) - \int sin (u) d u)

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

= - 3 \cdot \frac{1}{4} (u sin (u) - (- cos (u)))

u = 2 x

代回

= - 3 \cdot \frac{1}{4} (2 x sin (2 x) - (- cos (2 x)))

化简

- 3 \cdot \frac{1}{4} (2 x sin (2 x) - (- cos (2 x))) : - \frac{3}{4} (2 x sin (2 x) + cos (2 x))

= - \frac{3}{4} (2 x sin (2 x) + cos (2 x))

解答补常数

= - \frac{3}{4} (2 x sin (2 x) + cos (2 x)) + C

作图

流行的例子

integral of 2x^3-2/3 x^2+5x

\int 2 x^{3} - \frac{2}{3} x^{2} + 5 x d x

limit as x approaches 0 of (sin(10x))/x

x \to 0 lim (\frac{sin ( 10 x )}{x})

integral from 0 to 2 of (4x+3)/(x^4)

\int_{0}^{2} \frac{4 x + 3}{x ^{4}} d x

integral of (5x^3-7x^2+2x+1)

\int (5 x^{3} - 7 x^{2} + 2 x + 1) d x

derivative of 30+1/(x+1-3x^2)

\frac{d}{d x} (30 + \frac{1}{x + 1} - 3 x^{2})