   Chapter 5, Problem 6P ### 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

# If f ( x ) = ∫ 0 x x 2 sin ( t 2 )   d t , find f′(x).

To determine

The derivative of f(x).

Explanation

Given information:

Show the function as follows:

f(x)=0xx2sin(t2)dt (1)

Evaluate Equation (1) as follows:

f(x)=0xx2sin(t2)dt=x20xsin(t2)dt (2)

Differentiate Equation (2) by applying product rule:

f'(x)=x2(0xsin(t2)dt)+(x2)0xsin(t2)dt (3)

The Fundamental Theorem of Calculus, Part 1 is shown below:

g(x)=axf(t)dt axb

Here the continuous function on the interval [a,b] is f and the function is g

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 