   Chapter 3.9, Problem 5E

Chapter
Section
Textbook Problem

# Find the most general antiderivative of the function. (Check your answer by differentiation.) f ( x ) = x ( 12 x + 8 )

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=x(12x+8 )

4) Calculation:

The given function is fx=x(12x+8 )

Using distributive property gives

fx=12x2+8x

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,

12x2+12+1+8x1+11+1+C

Simplifying this,

12 x33+8 x22+C

123 x3+

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