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

1 Answer
Jul 4, 2017

f(x) = -3/2x^2 - x+5

Explanation:

Choose x=2 as lower limit if integration and pose:

f(x) = -3-int_2^x (3t+1)dt

Clearly:

f(2) = -3-int_2^2 (3t+1)dt = -3

and

(df)/dx = -3x-1

Now using the linearity of the integral:

f(x) = -3-int_2^x (3t+1)dt = -3 -3int_2^x tdt - int_2^xdt

f(x) = -3 -3 [t^2/2]_2^x - [t]_2^x

f(x) = -3 -3/2x^2 + 6 - x+2

f(x) = -3/2x^2 - x+5