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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Find the most general antiderivative of the function. (Check your answer by differentiation.)

f ( x ) = 1 5 2 x

To determine

To find: The most general antiderivative of the function f(x)=152x and check the determined antiderivative for the function f(x)=152x by differentiation.

Explanation

Given Data:

Write the given function as follows.

f(x)=152x

Formula used 1:

The antiderivative function for the function xn is xn+1n+1+C .

Here, C is the constant.

The antiderivative function for the function 1x is ln|x|+C .

Formula used 2:

Write the required differentiation formula to verify the answer as follows.

ddx(xn)=nxn1ddx(constant)=0ddx(ln|x|)=1x

Calculation:

Rewrite the function f(x)=152x as follows.

f(x)=15x02(1x) (1)

From the antiderivative function formulae, the antiderivative function for the function in equation (1) is written as follows.

F(x)=15(x0+10+1)2ln|x|+C=15x2ln|x|+C

Thus, the most general antiderivative of the function f(x)=152x is 15x2ln|x|+C_

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