Question

What is `f(x) = int e^(3x)-e^(2x)+xdx` if `f(0)=-2`?

Answer 1

`f(x)=1/3e^(3x)+1/2 (x^2-e^(2x)) -11/6`

Explanation:

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

`f(x)= inte^(3x)dx - inte^(2x)dx + intxdx`

`f(x)=1/3e^(3x)-1/2e^(2x)+1/2x^2 + C` ; where C`\equiv`constant

`f(x)=1/3e^(3x)+1/2 (x^2-e^(2x)) + C`

`f(0) = 1/3e^(3(0))-1/2e^(2(0))+cancel(1/2(0)^2) + C = -2`

`1/3e^(0)-1/2e^(0)+ C = -2`

`1/3(1)-1/2(1)+ C = -2`

`1/3 - 1/2 + C = -2`

`2/6 - 3/6 + C = -12/6`

`- 1/6 + C = -12/6`

`C = -11/6`

Hence:
`f(x)=1/3e^(3x)+1/2 (x^2-e^(2x)) -11/6`