Integration Rules

THEOREM 2

Let u and v be functions of x whose domains are an open interval I and suppose du and dv exist for every x in I.

(i)		du = u + C.
(ii)	Constant Rule	c du = c du.
(iii)	Sum Rule	du + dv = du + dv.
(iv)	Power Rule	where r is rational, r -1,and u > 0 on I.
(v)		sin u du = -cos u + C.
(vi)		cos u du = sin u + C.
(vii)		eu du = eu 4- C.
(viii)		1/u du = ln |u| + C (u 0).

Discussion

The Power Rule gives the integral of u^r when r -1, while Rule (viii) gives the integral of u^r when r = -1. When we put u =f(x) and v = g(x), the Constant and Sum Rules take the form

Constant Rule

c f(x) dx = c f(x) dx.

Sum Rule

(f (x) + g(x)) dx = f (x) dx + g(x) dx.

In the Constant and Sum Rules we are multiplying a family of functions by a constant and adding two families of functions. If we do either of these two things to families of functions differing only by a constant, we get another family of functions differing only by a constant. For example,

7(3x⁴ + C) = 21x⁴ + 7C = 21x⁴ + C'

is the family of all functions equal to 21x⁴ plus a constant. Similarly,

(3√x + C) + (5x - √x + D) = 5x + 2√x + (C + D) = 5x + 2√x + C

is the family of all functions equal to 5x + 2√x plus a constant.