   Chapter 5.3, Problem 16E ### 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

# Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. y = ∫ 0 x 4 cos 2 θ   d θ

To determine

The derivative of the function y using Part 1 of the Fundamental Theorem of Calculus.

Explanation

Given information:

y=0x4cos2θdθ

The integral function is y=0x4cos2θdθ (1)

The Part 1 of the Fundamental Theorem of Calculus is shown below.

F(x)=axf(t)dt

Consider that part 1 of the fundamental theorem is valid under following conditions:

• If f is a continuous function on the interval [a,b].
• The function is differentiable on the interval and F(x)=f(x).

Calculation:

Consider u=x4 (2)

Differentiate Equation (2) with respect to x.

dudx=4x3 (3)

Find the value of dydx using the relation given below.

dydx=dydududx (4)

Substitute y in Equation (4).

y=ddu0x4cos2θdθdudx (5)

Substitute u for x4 in Equation (5)

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