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受欢迎的微积分 >

integral of 5re^{r/2}

解答

\int 5 r e^{\frac{r}{2}} d r

解答

20 (\frac{1}{2} e^{\frac{r}{2}} r - e^{\frac{r}{2}}) + C

求解步骤

\int 5 r e^{\frac{r}{2}} d r

提出常数:

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

= 5 \cdot \int r e^{\frac{r}{2}} d r

使用换元积分法

= 5 \cdot \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

= 5 \cdot 4 \cdot \int e^{u} u d u

使用分布积分法

= 5 \cdot 4 (e^{u} u - \int e^{u} d u)

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

= 5 \cdot 4 (e^{u} u - e^{u})

u = \frac{r}{2}

代回

= 5 \cdot 4 (e^{\frac{r}{2}} \frac{r}{2} - e^{\frac{r}{2}})

化简

5 \cdot 4 (e^{\frac{r}{2}} \frac{r}{2} - e^{\frac{r}{2}}) : 20 (\frac{1}{2} e^{\frac{r}{2}} r - e^{\frac{r}{2}})

= 20 (\frac{1}{2} e^{\frac{r}{2}} r - e^{\frac{r}{2}})

解答补常数

= 20 (\frac{1}{2} e^{\frac{r}{2}} r - e^{\frac{r}{2}}) + C

作图

流行的例子

derivative f(x)=x^2ln(8-5x^2)

d er i v a t i v e f (x) = x^{2} ln (8 - 5 x^{2})

面积 y=x^3-10x,y=6x

a re a y = x^{3} - 10 x, y = 6 x

(dy)/(dx)=x^6y^{-5}

\frac{d y}{d x} = x^{6} y^{- 5}

integral from-5 to 5 of 1/(|x|^{2/3)}

\int_{- 5}^{5} \frac{1}{∣ x ∣ ^{\frac{2}{3}}} d x

derivative y=sqrt(arctan(x))

d er i v a t i v e y = arctan (x)