Question

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

Answer 1

`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`