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

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Answer 1 · 2017-07-04T23:17:31

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

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#