What is f(x) = int (x^2-2x)(e^x-3x) dx if f(1 ) = 4 ?

1 Answer
Feb 19, 2017

f(x) = (x^2-4x+4)e^x-3/4x^4+2x^3+11/4-e

Explanation:

Note that:

d/(dx) (x^2 e^x) = color(white)(XX) x^2 e^x + 2x e^x

d/(dx) (-4x e^x) = color(white)(XXX) - 4x e^x - 4e^x

d/(dx) (4e^x) = color(white)(XXXXXXXXXXX)4e^x

So:

d/(dx) (x^2-4x+4)e^x = (x^2-2x)e^x

Hence:

f(x) = int (x^2-2x)(e^x-3x) dx

color(white)(f(x)) = int (x^2-2x)e^x-3x^3+6x^2 dx

color(white)(f(x)) = (x^2-4x+4)e^x-3/4x^4+2x^3+C

So:

4 = f(1) = (1-4+4)e^1-3/4+2+C = e+5/4+C

Hence:

C = 4-(e+5/4) = 11/4-e

and:

f(x) = (x^2-4x+4)e^x-3/4x^4+2x^3+11/4-e