# The value of the integral function ∫ 0 1 d d x ( e arctan x ) d x . ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

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

#### Solutions

Chapter 5, Problem 8RE

(a)

To determine

## The value of the integral function ∫01ddx(earctanx) dx.

Expert Solution

The value of the integral function is eπ41_

### Explanation of Solution

Given information:

The integral function is 01ddx(earctanx)dx.

The region lies between x=0 and x=1.

Calculation:

Show the integral function as follows:

01ddx(earctanx)dx (1)

Apply the Fundamental Theorem of Calculus as follows:

F(x)=abG(x)dx

The expression to find the integral value by using Fundamental theorem of calculus as shown below.

01ddx(earctanx)dx=01ddx(earctanx)dx=(earctanx)01=(earctan1)(earctan0)=eπ4e0

01ddx(earctanx)dx=eπ41

Therefore, the value of the integral function is eπ41_.

(b)

To determine

### The value of the integral function ddx∫01(earctanx)dx.

Expert Solution

The value of the integral function is 0.

### Explanation of Solution

Given information:

The function as ddx01(earctanx)dx.

The region lies between x=0 and x=1.

Calculation:

Show the integral function as follows:

ddx01(earctanx)dx (2)

Use integral calculator to calculate the value as follows:

ddx01(earctanx)dx=ddx(1.591)=0

Therefore, the integral value of the function is 0. Since the definite integral is constant.

(c)

To determine

### The value of the integral function ddx∫0xddx(earctant) dt.

Expert Solution

The value of the integral function is earctanx_.

### Explanation of Solution

Given information:

The integral function is ddx0xddx(earctant)dt.

Calculation:

Show the integral function as follows:

ddx0xddx(earctant)dt (3)

The expression to find the integral value by using Fundamental theorem of calculus as shown below.

ddx0x(earctant)dt=earctanx

Therefore, the value of the integral function is earctanx_.

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