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受欢迎的微积分 >

integral of x/2 cos(2x)

解答

\int \frac{x}{2} cos (2 x) d x

解答

\frac{1}{8} (2 x sin (2 x) + cos (2 x)) + C

求解步骤

\int \frac{x}{2} cos (2 x) d x

提出常数:

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

= \frac{1}{2} \cdot \int x cos (2 x) d x

使用换元积分法

= \frac{1}{2} \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

= \frac{1}{2} \cdot \frac{1}{4} \cdot \int u cos (u) d u

使用分布积分法

= \frac{1}{2} \cdot \frac{1}{4} (u sin (u) - \int sin (u) d u)

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

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

u = 2 x

代回

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

化简

\frac{1}{2} \cdot \frac{1}{4} (2 x sin (2 x) - (- cos (2 x))) : \frac{1}{8} (2 x sin (2 x) + cos (2 x))

= \frac{1}{8} (2 x sin (2 x) + cos (2 x))

解答补常数

= \frac{1}{8} (2 x sin (2 x) + cos (2 x)) + C

作图

流行的例子

integral of 1/4 sin(x)

\int \frac{1}{4} sin (x) d x

integral of ((x+2)^3)/(x^2)

\int \frac{( x + 2 ) ^{3}}{x ^{2}} d x

integral of 0.75x^2

\int 0.75 x^{2} d x

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

\int \frac{1 + 3 x + 7 x ^{2} - 2 x ^{3}}{x ^{2}} d x

integral of 0.75(1-x^2)

\int 0.75 (1 - x^{2}) d x