   Chapter 3.9, Problem 4E

Chapter
Section
Textbook Problem

# Find the most general antiderivative of the function. (Check your answer by differentiation.) f ( x ) = 6 x 5 − 8 x 4 − 9 x 2

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=6x5-8x4-9x2

4) Calculation:

Here, fx=6x5-8x4-9x2

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,

6x5+15+1-8x4+14+1-9x2+12+1+C

Simplifying this,

6 x66-8 x55-9x33+C

66 x6-85x5-93 x3+C

Which simplifies to,

x6-85x5-3 x

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