   Chapter 4.9, Problem 14E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### 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 ( t ) = 3 t 4 − t 3 + 6 t 2 t 4

To determine

To find: The most general antiderivative of the function f(t)=3t4t3+6t2t4 and check the determined antiderivative for the function f(t)=3t4t3+6t2t4 by differentiation.

Explanation

Given Data:

Write the given function as follows.

f(t)=3t4t3+6t2t4

Formula used 1:

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

Here, C is the constant.

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

Formula used 2:

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

ddt(tn)=ntn1ddt(constant)=0ddt(ln|t|)=1t

Calculation:

Rewrite the function f(t)=3t4t3+6t2t4 as follows.

f(t)=3t4t4t3t4+6t2t4

f(t)=3t01t+6t2 (1)

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

F(t)=3(t0+10+1)ln|t|+6(t2+12+1)+C=3tln|t|+6(t11)+C=3tln|t|6t+C

Thus, the most general antiderivative of the function f(t)=3t4t3+6t2t4 is 3tln|t|6t+C_ .

Here, the determined antiderivative for the function f(t)=3t4t3+6t2t4 is 3tln|t|6t+C

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