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Section 5.2 : Computing Indefinite Integrals
22. Determine \(f\left( x \right)\) given that \(f'\left( x \right) = 12{x^2} - 4x\) and \(f\left( { - 3} \right) = 17\).
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Hint : We know that integration is simply asking what function we differentiated to get the integrand and so we should be able to use this idea to arrive at a general formula for the function.
Recall from the notes in this section that we saw,
\[f\left( x \right) = \int{{f'\left( x \right)\,\,dx}}\]and so to arrive at a general formula for \(f\left( x \right)\) all we need to do is integrate the derivative that we’ve been given in the problem statement.
\[f\left( x \right) = \int{{12{x^2} - 4x\,\,dx}} = 4{x^3} - 2{x^2} + c\]Don’t forget the “+c”!
Hint : To determine the value of the constant of integration, \(c\), we have the value of the function at \(x = - 3\).
Because we have the condition that \(f\left( { - 3} \right) = 17\) we can just plug \(x = - 3\) into our answer from the previous step, set the result equal to 17 and solve the resulting equation for \(c\).
Doing this gives,
\[17 = f\left( { - 3} \right) = - 126 + c\hspace{0.5in} \Rightarrow \hspace{0.5in}c = 143\]The function is then,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{f\left( x \right) = 4{x^3} - 2{x^2} + 143}}\]