   Chapter 3.9, Problem 14E

Chapter
Section
Textbook Problem

# Find the most general antiderivative of the function. (Check your answer by differentiation.) g ( x ) = 5 − 4 x 3 + 2 x 6 x 6

To determine

To find:

The most general antiderivative of the given function.

Explanation

1) Concept:

If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is,Fx+C where C  is an arbitrary constant.

Definition:

A function F  is called an antiderivative of f on an interval I if

F'x=fx for all x in I.

2) Formula:

Power rule of antiderivative

ddx xn+1n+1=xn

3) Given:

gx=5-4x3+21x6x6

4) Calculation:

Given function gx=5-4x3+ 21 x 6x6

Therefore, the function can also be rewritten as gx=5x-6-4x-3+2

This function is not defined at x=0

That is, the function gx=5-4x3+21x6x6  has domain -, 00,

To find the most general antiderivative of f, use the power rule of antiderivative.

Power rule of antiderivative

ddx xn+1n+1=xn

Using power rule on gx= 5x-6-4x-3+2  gives

5 x-6+1-6+1 -4 x-3+1-3+1 +2 x0+10+1+C

5 x-5 -5 +4 x-2-2+2 x1+C

5-5  x-5-4-2  x-2 +2 x+C

Which simplifies to

- x-5+2 x-2+2x+C

Which can also be written as

-1x5+2x2+2x+C

Given function, gx=5x-6-4x-3+2 is not defined at x=0

Thus theorem 1 tells us only that the general antiderivative of f is -x-5+2 x-2+2x+C  on any interval that does not contain 0

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