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

1 Answer
Mar 4, 2018

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\equivconstant

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