Page 227 - Calculus Demystified

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SOLUTION CHAPTER 7 Methods of Integration
We write
π/4 π/4

4 2 2
sec xdx = sec x · sec xdx
0 0
π/4

2
2
= (tan x + 1) sec xdx.
0
2
Letting u = tan x and du = sec xdx then gives the integral
1 3
1
2 u
u + 1 du = + u
0 3 0
4
= .
3
You Try It: Calculate the integral
2π
4
6
sin x cos xdx.
π
Further techniques in the evaluation of trigonometric integrals will be explored
in the exercises.

Exercises

1. Use integration by parts to evaluate each of the following indefinite
integrals.

2
(a) log xdx

(b) x · e 3x dx

2
(c) x cos xdx

(d) t sin 3t cos 3tdt

(e) cos y ln(sin y) dy

2 4x
(f) x e dx