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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 ) = 2 x 2 + 5 x 2 + 1

To determine

To find: The most general antiderivative of the function f(x)=2x2+5x2+1 and check the determined antiderivative for the function f(x)=2x2+5x2+1 by differentiation.

Explanation

Given Data:

Write the given function as follows.

f(x)=2x2+5x2+1

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 1x2+1 is tan1x+C .

Formula used 2:

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

ddx(xn)=nxn1ddx(tan1x)=1x2+1ddx(constant)=0

Calculation:

Rewrite the function f(x)=2x2+5x2+1 as follows.

f(x)=2x2x2+1+5x2+1=2(x2x2+1)+5(1x2+1)=2(x2+11x2+1)+5(1x2+1)=2(x2+1x2+11x2+1)+5(1x2+1)

f(x)=2(11x2+1)+5(1x2+1)

f(x)=2(x01x2+1)+5(1x2+1) (1)

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

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