Calculus: Analytic Geometry and Calculus, with Vectors

524 Exponential and logarithmic functions

and is quite near 1 even when n is as small as 4 or 3. Thus, even when n is quite

small, the number 1/\ is a very good approximation to the probability of

finding exactly n heads and exactly n tails when 2n coins are tossed. This

astonishing result has very significant consequences, and persons who comprehend

it may say that mathematics was never so beautiful before, and mathematics

really is the Queen of the Sciences.

7 Derive the formula

(1) I ersx sin bx dx =

e°x(a sinabxl-bb cos bx) +

in the following way. Set

to obtain

(2)

Then set

to obtain

(3)

u=ell, dv=sinbxdx

a

J

ellsinbx dx = -

e

b

s x

+ bfe0xcosbx dx.

u = sin bx, de = ell dx

1

e °, sin bx dx =e°x sin bx-b jJ

a a ell cos bx dx.

Then combine (2) and (3) to obtain (1). Now derive (1) in the following way.

Let I denote the left members of (1), (2), and (3). Set

u = eax, de = cos bx dx

to put (2) in the form

(4)

ell cos bx + a eax sin bx-a I

b bL b b J

Then solve (4) for I and obtain (1).

8 Derive the formula

f ell cos bx dx =

e°x(b sin bx + 2 cos bx) +

c

..9 Sketch graphs of y = ex and y = log x in the same figure. Let Rl be

the region consisting of points (x,y) for which x < 0 and 0 <y 5 Cr. This

unbounded region Rl is said to be bounded by (or to have boundaries) thex axis,

the y axis, and the graph of y = ex. Let R2 be the region consisting of points

(x,y) for which x = 0 and y 5 0 together with points (x,y) for which 0 <x 5 1

and log x 5 y 5 0. This region R2 is said to be bounded by the x axis, the y

axis, and the graph of y = log x. Observe that R, and R2 are congruent regions.

Use the fact that Rl and R2 possess areas IR1J and JR21 for which

Ad = lim f h e x dx, JR21 = lim fhI (-log x) dx

to show that IRIJ = IR2I = 1.