What is f(x) = int 3x^3+xe^(x-2)+e^x dxf(x)=∫3x3+xex−2+exdx if f(2 ) = 4 f(2)=4?

1 Answer
Gió
Dec 27, 2016

I found: f(x)=3/4x^4+e^(x-2)(x-1)+e^x-9-e^2f(x)=34x4+ex−2(x−1)+ex−9−e2

Explanation:

Let us first solve our integral using also integration by parts on xe^(x-2)xex−2:
f(x)=int3x^3dx+intxe^(x-2)dx+inte^xdxf(x)=∫3x3dx+∫xex−2dx+∫exdx
f(x)=3x^4/4+xe^(x-2)-inte^(x-2)dx+e^xf(x)=3x44+xex−2−∫ex−2dx+ex
f(x)=3/4x^4+xe^(x-2)-e^(x-2)+e^x+cf(x)=34x4+xex−2−ex−2+ex+c
f(x)=3/4x^4+e^(x-2)(x-1)+e^x+cf(x)=34x4+ex−2(x−1)+ex+c
now we evaluate it at x=2x=2 to get:
f(2)=3/4(2^4)+e^(2-2)(2-1)+e^2+cf(2)=34(24)+e2−2(2−1)+e2+c
f(2)=12+1+e^2+cf(2)=12+1+e2+c
f(2)=13+e^2+cf(2)=13+e2+c

we use the fact that f(2)=4f(2)=4 to find cc and write:
13+e^2+c=413+e2+c=4
so that rearranging:
c=-9-e^2c=−9−e2

and our function will be:

f(x)=3/4x^4+e^(x-2)(x-1)+e^x-9-e^2f(x)=34x4+ex−2(x−1)+ex−9−e2

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