Question

What is `f(x) = int 3x^3+xe^(x-2) dx` if `f(2 ) = 3`?

Answer 1

`f(x)=3/4x^4+(x-1)e^(x-2)-9-e.`

Explanation:

`f(x)=int(3x^3+xe^(x-2))dx.`

`=int3x^3dx+intxe^(x-2)dx`

`=3(x^(3+1)/(3+1))+I, where, I=intxe^(x-2)dx,`

`:. f(x)=3/4x^4+I,................(ast).`

To evaluate `I,` we use the following Method of Integration

by Parts (IBP).

`(IBP) : intuvdx=uintvdx-int[(du)/dx*intvdx]dx.`

We take, `u=x rArr (du)/dx=1, &, v=e^(x-2) rArr intvdx=e^(x-2).`

`:. I=xe^(x-2)-int{1*e^(x-2)}dx,`

`:. I=xe^(x-2)-e^(x-2)=(x-1)e^(x-2).............(star).`

`(ast) and (star) rArr f(x)=3/4x^4+(x-1)e^(x-2)+C.`

To determine `C," we use the given cond. : "f(2)=3.`

`rArr 3/4(2)^4+(2-1)e^(3-2)+C=3 rArr C=-9-e`

Therefore, `f(x)=3/4x^4+(x-1)e^(x-2)-9-e.`

Enjoy Maths.!