Page 180 - Calculus Demystified

CHAPTER 6
Similarly, Transcendental Functions 167
d 1
(log x) =
8
dx x · ln 8
d 1 d
(log (x · cos x)) = · (x · cos x)
4
dx (x · cos x) · ln 4 dx
cos x + (x · (− sin x))
= .
(x · cos x) · ln 4
EXAMPLE 6.26
Integrate

2
3 cot x · (− csc x) dx.

SOLUTION
2

Forclaritywesetϕ(x) = cot x,ϕ (x) =− csc x.Thenourintegralbecomes
1

3 ϕ(x) · ϕ (x) dx = · 3 ϕ(x) · ϕ (x) · ln 3 dx
ln 3

1
= · 3 ϕ(x) + C.
ln 3
Resubstituting the expression for ϕ(x) now gives that
1
2
3 cot x · (− csc x) dx = · 3 cot x + C.
ln 3
3
You Try It: Evaluate (log (x )/x) dx.
6
You Try It: Calculate the integral
2
x
x · 3 dx.

Our new ideas about arbitrary exponents and bases now allow us to formulate a
general result about derivatives of powers:
For any real exponent a we have
d a a−1
x = a · x .
dx
EXAMPLE 6.27
√
−π
e
Calculate the derivative of x , x 3 , x .