   Chapter 7.4, Problem 62E

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. ∫ π / 3 π / 2 1 1 + sin x − cos x d x

To determine

To evaluate the integralπ3π211+sinxcosxdx

Explanation

Calculation: We substitute t=tanx2,π<x<π then by exercise 59

cosx=1t21+t2,sinx=2t1+t2,dx=21+t2dt

Finding the limits of integration, we get

t=tanπ32=tanπ6=13

t=tanπ22=tanπ4=1

Now substitute everything into the integral

π3π211+sinxcosxdx=13111+2t1+t21t21+t2.21+t2dt

=13111+2t1+t21+t2.21+t2dt

=13121+t2+2t1+t2dt

=13122t2+2tdt

=1311t2+tdt

Partial fraction decomposition is given by

1t2+t<

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