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

1 Answer
Jun 12, 2017

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.!