Page 133 - Calculus Demystified

120
4.5 Rules of Integration CHAPTER 4 The Integral
We have been using various rules of integration without enunciating them explicitly.
It is well to have them recorded for future reference.

4.5.1 LINEAR PROPERTIES

I. If f, g are continuous functions on [a, b] then

b b b
f(x) + g(x) dx = f(x) dx + g(x) dx.
a a a
II. If f is a continuous function on [a, b] and c is a constant then

b b
cf (x) dx = c f(x) dx.
a a

4.5.2 ADDITIVITY

III. If f is a continuous on [a, c] and a< b < c then
b c c

f(x) dx + f(x) dx = f(x) dx.
a b a
You Try It: Calculate
3

2
3
4x − 3x + 7x − 12 cos xdx.
1
Exercises

1. Calculate each of the following antiderivatives:
2
(a) Antiderivative of x − sin x
4
(b) Antiderivative of e 3x + x − 2
2
(c) Antiderivative of 3t + (ln t/t)
(d) Antiderivative of tan x − cos x + sin 3x
(e) Antiderivative of cos 3x + sin 4x + 1
(f) Antiderivative of (cos x) · e sin x