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Section 5-1 : Indefinite Integrals

6. Determine \(f\left( x \right)\) given that \(f'\left( x \right) = 6{x^8} - 20{x^4} + {x^2} + 9\).

Hint : Remember that all indefinite integrals are asking us to do is “undo” a differentiation.
Show Solution

We know that indefinite integrals are asking us to undo a differentiation to so all we are really being asked to do here is evaluate the following indefinite integral.

\[f\left( x \right) = \int{{f'\left( x \right)\,\,dx}} = \int{{6{x^8} - 20{x^4} + {x^2} + 9\,dx}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{2}{3}{x^9} - 4{x^5} + \frac{1}{3}{x^3} + 9x + c}}\]

Don’t forget the “+c”! Remember that the original function may have had a constant on it and the “+c” is there to remind us of that.