   Chapter 5.2, Problem 67E ### 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

# Which of the integrals ∫ 1 2 arctan x   d x , ∫ 1 2 arctan x   d x , and ∫ 1 2 arctan ( sin x ) d x has the largest value? Why?

To determine

To find: The larger value of integral among the integral function 12arctanxdx12arctanxdx and 12arctan(sinx)dx.

To explain: The reason for obtaining largest value of the integral.

Explanation

The functions f(x)=arctanx, g(x)=arctanx are increasing function. The function h(x)=arctan(sinx) first increases and then decreases within limit [1,2].

Compare the table 1, table 2 and table 3.

f(x)g(x)>h(x)arctanxarctanx>arctan(sinx)

Given information:

The integral function 12arctanxdx,12arctanxdx and 12arctan(sinx)dx.

within limits [1,2]

Calculation:

Consider the value of function.

f(x)=arctanx (1)

g(x)=arctanx (2)

h(x)=arctan(sinx) (3)

Calculate the value of f(x) within limits [1,2] with increment of 0.2, using Equation (1).

Calculate the value of f(1), using Equation (1).

Substitute 1 for x in Equation (1).

f(1)=arctan(1)=0.78540

Similarly, calculate and tabulate the remaining values as in table 1.

Table 1

 x f(x)=arctanx 1 0.78540 1.2 0.87606 1.4 0.95055 1.6 1.01220 1.8 1.06370 2 1.10715

Refer Table 1:

The function f(x)=arctanx is a increasing function within limits [1,2]

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