Page 167 - Calculus Demystified

CHAPTER 6
154
Transcendental Functions
For clarity we take ϕ(x) = sin x, ϕ (x) = cos x. Then the integral becomes

ϕ (x)
dx = ln |ϕ(x)|+ C.
ϕ(x)
Resubstituting the expression for ϕ yields the solution:

cot xdx = ln | sin x|+ C.

6.2 Exponential Basics

Examine Fig. 6.4, which shows the graph of the function

f(x) = ln x, x > 0.

Fig. 6.4

As we observed in Section 1, the function f takes on all real values. We already
have noticed that, since
d 1
ln x = > 0,
dx x
the function ln x is increasing. As a result,

ln :{x : x> 0}→ R

is one-to-one and onto. Hence the natural logarithm function has an inverse.