   Chapter 3.9, Problem 8E

Chapter
Section
Textbook Problem

# Find the most general antiderivative of the function. (Check your answer by differentiation.) f ( x ) = x 3.4 − 2 x 2 − 1

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=x3.4-2 x2 -1

4) Calculation:

The given function is fx=x3.4-2 x2 -1

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,

x3.4+1 3.4+1 -2 x2-1+12-1+1+C

x4.44.4-2 x22+C

14.4 x4.4-22  x2+C

14.4 can be written in the fraction form as 1044

Which can be reduced further to 522

22  can be written as 2·2 2  which gives 2

Thus, the expression can be rewritten as

522 x4

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