Find the most general antiderivative of the function f(x) = 5e^x + 8 sec^2 x on the interval...


Find the most general antiderivative of the function {eq}f(x) = 5e^x + 8 sec^2 x {/eq} on the interval {eq}(\frac{n}{\pi} - \frac{\pi}{2}, \frac {n}{\pi}+\frac{\pi}{2}) {/eq}


The opposite of a derivative is an antiderivative. We can reverse our various differentiation techniques to find the general form of an antiderivative for many functions. Each time we find an antiderivative, we're defining a family of functions, as there are an infinite amount of antiderivatives for any specific function.

Answer and Explanation:

The most general antiderivative of any function will have an unknown constant term at the end. We can find the antiderivative term by term of this expression and then add a constant onto the end. To do so, we need to remember two important definitions for differentiation:

{eq}\frac{d}{dx} e^x = e^x\\ \frac{d}{dx} \tan x = \sec^2x {/eq}

By reversing these definitions, tacking on the coefficients, adding the terms together, and adding our constant term to the end, we can find the most general antiderivative of this function to be the following:

{eq}F(x) = 5e^x + 8 \tan x + c {/eq}

Learn more about this topic:

Antiderivative: Rules, Formula & Examples

from Calculus: Help and Review

Chapter 8 / Lesson 12

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