   Chapter 7.4, Problem 63E

Chapter
Section
Textbook Problem

# Use the substitution in Exercise 59 to transform the integrand into a rational function of t and then evaluate the integral. ∫ 0 π / 2 sin 2 x 2 + cos x d x

To determine

To evaluate the integral 0π2sin2x2+cosxdx

Explanation

Calculation:

0π2sin2x2+cosxdx=0π22sinxcosx2+cosxdx

Let tanx2=u, then

12sec2x2dx=du12(1+tan2x2)dx=du12(1+u2)dx=dudx=21+u2du

sinx=2tanx21+tan2x2=2u1+u2,cosx=1tan2x21+tan2x2=1u21+u2

Therefore

0π2sin2x2+cosxdx=0π222u1+u2.1u21+u22+1u21+u2.21+u2du=801u(1u2)(1+u2)2(3+u2)du

Resolve into partial factor decomposition, we get

u(1u2)(1+u2)2(3+u2)=Au+B1+u2+Cu+D(1+u2)2+Eu+F3+u2u(1u2)=(Au+B)(1+u2)(3+u2)+(Cu+D)(3+u2)+(Eu+F)(1+u2)2

Upon solving, we get the values

A=1,B=0,C=1,D=0,E=1,F=0

u(1u2)(1+u2)2(3+u2

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