   Chapter 3.9, Problem 13E

Chapter
Section
Textbook Problem

# Find the most general antiderivative of the function. (Check your answer by differentiation.) f ( x ) = 10 x 9

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:

fx=10x9

4) Calculation:

The given function is fx=10x9

Recall that 10x9  can also be written as 10 x-9

Therefore, the function can also be rewritten as fx=10 x-9

This function is not defined at x=0

That is, the function fx=10 x-9  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

Which gives,

10 x-9+1-9+1 +C

10 x-8-8  +C

10-8  x-8 +C

5-4 x-8+C

Which simplifies to,

- 54x8+C

The given function fx=10x9= 10x-9 is not defined at x=0

Thus, theorem 1 tells us only that the general antiderivative of f is -54x8+ C   on any interval that does not contain 0

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