Question

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

Answer 1

`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`