BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 5.4, Problem 11E
To determine

To find: use part 1 of the fundamental theorem of calculus to find the derivative of the function.

Expert Solution

Answer to Problem 11E

The derivative of the function is F(x)=-1+sec x .

Explanation of Solution

Given information: The function is F(x)=xπ1+sec t dt

Let’s remind of the fundamental theorem of calculus part 1:

The fundamental theorem of calculus part 1: If f is continuous on [a,b] then the function of g defined by

  g(x)=axf(t) dt where axb is continuous on [a,b] and differentiable on (a,b) and g(x)=f(x) .

First, we will use the properties of definite integral to match the form in fundamental theorem of calculus part 1.

So, xπ1+sec t dt= -πx1+sec t dt where F(t)=-F(t)

Hence,

  F(x)=-πx1+sec t dt

Now given, F(x)=-πx1+sec t dt

Differentiation with respect to x,

  d F(x)dx=ddx[πx1+sec t dt]

  F(x)=-1+sec x [ [ where g(x)=f(x)]

Thus, F(x)=-1+sec x

Hence, The derivative of the function is F(x)=-1+sec x .

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