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受欢迎的微积分 >

integral of xe^{x/2}

解答

\int x e^{\frac{x}{2}} d x

解答

4 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}}) + C

求解步骤

\int x e^{\frac{x}{2}} d x

使用换元积分法

= \int 4 e^{u} u d u

提出常数:

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

= 4 \cdot \int e^{u} u d u

使用分布积分法

= 4 (e^{u} u - \int e^{u} d u)

\int e^{u} d u = e^{u}

= 4 (e^{u} u - e^{u})

u = \frac{x}{2}

代回

= 4 (e^{\frac{x}{2}} \frac{x}{2} - e^{\frac{x}{2}})

化简

4 (e^{\frac{x}{2}} \frac{x}{2} - e^{\frac{x}{2}}) : 4 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}})

= 4 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}})

解答补常数

= 4 (\frac{1}{2} e^{\frac{x}{2}} x - e^{\frac{x}{2}}) + C

作图

流行的例子

斜率 (8.4)(5.2)

s l o p e (8.4) (5.2)

integral of-(3000)/(x^2)

\int - \frac{3000}{x ^{2}} d x

integral of (10x)/(sqrt(4x^2-4))

\int \frac{10 x}{4 x ^{2} - 4} d x

inverselaplace ((s+3))/((s+7))

in v erse l a pl a ce \frac{( s + 3 )}{( s + 7 )}

y^'+(2y)/t =(2cos(t^2))/t

y^{^{'}} + \frac{2 y}{t} = \frac{2 cos ( t ^{2} )}{t}