Chapter 5, Problem 8RE

(a)

To determine

The value of the integral function ${\displaystyle \underset{0}{\overset{1}{\int }}\frac{d}{dx}\left(e^{\arctan x}\right)}dx$.

Answer to Problem 8RE

The value of the integral function is $\underset{\_}{e^{\frac{\pi }{4}}-1}$

Explanation of Solution

Given information:

The integral function is ${\displaystyle \underset{0}{\overset{1}{\int }}\frac{d}{dx}\left(e^{\arctan x}\right)}dx$.

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

Calculation:

Show the integral function as follows:

${\displaystyle \underset{0}{\overset{1}{\int }}\frac{d}{dx}\left(e^{\arctan x}\right)}dx$ (1)

Apply the Fundamental Theorem of Calculus as follows:

$F\left(x\right)={\displaystyle \underset{a}{\overset{b}{\int }}{G}^{\prime }\left(x\right)dx}$

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

$\begin{array}{c}{\displaystyle \underset{0}{\overset{1}{\int }}\frac{d}{dx}\left(e^{\arctan x}\right)dx}={\displaystyle \underset{0}{\overset{1}{\int }}\frac{d}{dx}\left(e^{\arctan x}\right)}dx\\ ={\left(e^{\arctan x}\right)}_0^1\\ =\left(e^{\arctan 1}\right)-\left(e^{\arctan 0}\right)\\ =e^{\frac{\pi }{4}}-e^0\end{array}$

${\displaystyle \underset{0}{\overset{1}{\int }}\frac{d}{dx}\left(e^{\arctan x}\right)dx}=e^{\frac{\pi }{4}}-1$

Therefore, the value of the integral function is $\underset{\_}{e^{\frac{\pi }{4}}-1}$.

(b)

To determine

The value of the integral function $\frac{d}{dx}{\displaystyle \underset{0}{\overset{1}{\int }}\left(e^{\arctan x}\right)dx}$.

Answer to Problem 8RE

The value of the integral function is 0.

Explanation of Solution

Given information:

The function as $\frac{d}{dx}{\displaystyle \underset{0}{\overset{1}{\int }}\left(e^{\arctan x}\right)dx}$.

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

Calculation:

Show the integral function as follows:

$\frac{d}{dx}{\displaystyle \underset{0}{\overset{1}{\int }}\left(e^{\arctan x}\right)dx}$ (2)

Use integral calculator to calculate the value as follows:

$\begin{array}{c}\frac{d}{dx}{\displaystyle \underset{0}{\overset{1}{\int }}\left(e^{\arctan x}\right)dx}=\frac{d}{dx}\left(1.591\right)\\ =0\end{array}$

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

(c)

To determine

The value of the integral function $\frac{d}{dx}{\displaystyle \underset{0}{\overset{x}{\int }}\frac{d}{dx}\left(e^{\arctan t}\right)}dt$.

Answer to Problem 8RE

The value of the integral function is $\underset{\_}{e^{\arctan x}}$.

Explanation of Solution

Given information:

The integral function is $\frac{d}{dx}{\displaystyle \underset{0}{\overset{x}{\int }}\frac{d}{dx}\left(e^{\arctan t}\right)}dt$.

Calculation:

Show the integral function as follows:

$\frac{d}{dx}{\displaystyle \underset{0}{\overset{x}{\int }}\frac{d}{dx}\left(e^{\arctan t}\right)}dt$ (3)

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

$\frac{d}{dx}{\displaystyle \underset{0}{\overset{x}{\int }}\left(e^{\arctan t}\right)}dt=e^{\arctan x}$

Therefore, the value of the integral function is $\underset{\_}{e^{\arctan x}}$.