differentiate e^x².hence determine a,b,c for which ∫limit 1to 2 x³e^x² dx=a/b(e^c)

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violy | High School Teacher | (Level 1) Associate Educator

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`d(e^u) = e^udu`

So for derivative of `e^(x^2)` , `e^(x^2) = 2xe^(x^2)`

`int_1^2x^3e^(x^2)dx`  

Let set `u = x^2, du = 2xdx`

and `dv = xe^(x^2)dx , v = (1/2)e^(x^2)`

Appy Integration by parts formula: 

`intudv = uv -intvdu`  

`int_1^2x^3e^(x^2)dx = (1/2)x^2e^(x^2) - intxe^(x^2)dx`

                       `= (1/2)x^2e^(x^2) - (1/2)e^(x^2) ` 

                        `= e^(x^2)((1/2)x^2 - (1/2))`` `

Plug-in the upper limit and lower limit. 

`[e^(2^2)((1/2)(2^2) - (1/2))]- [e^(1^2)((1/2)(1^2) - (1/2)] = (3/2)e^4`

So, a = 3, b = 2, and c = 4

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pramodpandey | College Teacher | (Level 3) Valedictorian

Posted on

let

`t=e^(x^2)`

differentiate t w.r.t. x

`dt/dx=e^(x^2)(d/dx)(x^2)=2xe^(x^2)`      (i)``

`y=x^2`

`dy=2xdx , `

`dy/2=xdx`

`x=1 ,y=1 `

`x=2 ,y=2^2=4`

 

`x^3e^(x^2)=x^2 x e^(x^2)`

`int_1^2x^3e^(x^2)dx=int_1^2x^2e^(x^2)xdx=int_1^4ye^y(dy/2)`

`=(1/2)int_1^4ye^ydy`

,integrating by parts with respect to y ,by considering y as first function and `e^y`  second function.

`=(1/2){ye^y-e^y}_1^4`

`=(1/2){(4e^4-e^4)-(1e^1-e^1)}`

`=(1/2)(3e^4)`

`=(3/2)e^4`

`a=3,b=2,` and `c=4`

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