   Chapter 8, Problem 5PS

Chapter
Section
Textbook Problem

Area Use the substitution u = tan x 2 vto find the area of the shaded region under the graph of y = 1 2 + cos x for 0 ≤ x ≤ π / 2 (see figure). To determine

To calculate: The area of the shaded region in the given graph by using substitution of u=tanx2. Explanation

Given:

The function for the graph is y=12+cosx for [0,π2].

The given region is as shown below.

Formula used:

The trigonometric formula

cos2A=1tan2A1+tan2A

ddx(tanx)=sec2x

sec2x=tan2x+1

Calculation:

The value is taken as u=tanx2.

Differentiate the above given equation with respect to x.

ddxu=ddx(tanx2)=12sec2x2

Multiplying dx on both sides of the above equation, we get

du=12sec2x2dxdu=1+tan2x22dxdx=21+tan2x2dudx=21+u2du

As the function is y=12+cosx, use the formula

cosx=1tan2x21+tan2x2=1u2<

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