Page 208 - Calculus Demystified

Transcendental Functions
CHAPTER 6
2
2
2
(b) x ·ln xdx [Hint: Guess p(x)·ln x +q(x) ln x +r(x), p, q, r 195
polynomials.]
e ln x
(c) dx
1 x
2 e x

(d) x dx
1 e + 1
5. Use the technique of logarithmic differentiation to calculate the derivative
of each of the following functions.
2
x + 1
3
(a) x ·
3
x − x
3
sin x · (x + x)
(b)
2
x (x + 1)
2
3 4
2
(c) (x + x ) · (x + x) −3 · (x − 1)
x · cos x
(d)
ln x · e x
6. There are 5 grams of a certain radioactive substance present at noon on
January 10 and 3 grams present at noon on February 10. How much will be
present at noon on March 10?
7. A petri dish has 10,000 bacteria present at 10:00 a.m. and 15,000 present at
1:00 p.m. How many bacteria will there be at 2:00 p.m.?
8. A sum of $1000 is deposited on January 1, 2005 at 6% annual interest,
compounded continuously. All interest is re-invested. How much money
will be in the account on January 1, 2009?
9. Calculate these derivatives.
d −1 x
(a) Sin (x · e )
dx

d x
(b) Tan −1
dx x + 1
d
2
(c) Tan −1 [ln(x + x)]
dx
d −1
(d) Sec (tan x)
dx
10. Calculate each of these integrals.

2x
(a) xdx
1 + x 4