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8.4 Improper Integrals
AP Calculus BCAP Calculus BC
8.4 Improper Integrals
One of the great characteristics of mathematics is that mathematicians are constantly finding ways to get around the rules, or to bend the rules, or just plain ignore them.
Improper integrals is a technique to use when an interval is not finite, and when an integrand is not continuous.
Until now we have been finding integrals of continuous functions over closed intervals.
Sometimes we can find integrals for functions where the function or the limits are infinite. These are called improper integrals.
First, here are the definitions of Improper Integrals with Infinite Solutions:
1) If is continuous on , , then
limb
a ab
f x a
f x dx f x dx
2) If is continuous on , , then
limb b
aa
f x b
f x dx f x dx
If the Limit is finite, then the Improper Integral converges.
If the Limit fails, then it diverges
3) If is continuous on , , thenc
c
f x
f x dx f x dx f x dx
If both Improper Integrals
converge, then so does .f x dx
Express the improper integral as a limit of definite integrals and evaluate the integral.
22
2
1
xdx
x
First, rewrite and split up the integral.
22
2
1
xdx
x
22
0 2
1
xdx
x
220
2
1
xdx
x
0
22
2lim
1aa
xdx
x
20 2
2lim
1
b
b
xdx
x
2
0
1lim
1
b
b x
2 20
1 1lim
1 1b b
1
The antiderivative of 20 2
2
1
b xdx
x
= 0 as b → ∞
Express the improper integral as a limit of definite integrals and evaluate the integral.
22
2
1
xdx
x
Now, by analyzing the integrand, the denominator is always positive for all x. [i.e., (x2 + 1)2 is always positive.]
So, 22
2
1
x
x is positive when x > 0 and negative when x < 0.
Therefore,
0
22
21
1
xdx
x
because x < 0
and, 22
2
1
xdx
x
= −1 + 1 = 0
2
0
1lim
1
b
b x
2 2
1 1lim
1 0 1b b
1
Evaluate the following integral or state that it diverges.1
xxe dx
Use Integration by Parts u dv uv v du u xdu dx
xdv e dxxv e
1
xxe dx 1
bxx e 1
b xe dx 1
bx xx e e
1
1bxe x 11 2be b e
1 2lim
bb
b
e e
= 0
2
e
Improper Integrals with Infinite Discontinuities
Integrals of functions that become infinite at a point within the interval of integration are improper integrals.
1) If is continuous on a, b , then
limb b
a cc a
f x
f x dx f x dx
2) If is continuous on , , then
limb c
a ac b
f x a b
f x dx f x dx
3) If is continuous on , c, b , thenb c b
a a c
f x a c
f x dx f x dx f x dx
Another type of improper integral arises when the integrand has a vertical asymptote. And that’s also a point of infinite discontinuity either at a limit of integration or at some other point between the limits of integration.
Finite Limit Converges
Limit Fails Diverges
Both Limits Finite Converges
Example 1:
1
0
1
1
xdx
x
The function is
undefined at x = 1 .
Since x = 1 is an asymptote, the function has no maximum.
Can we find the area under an infinitely high curve?
We could define this integral as:
01
1lim
1
b
b
xdx
x
(left hand limit)
We must approach the limit from inside the interval.
01
1lim
1
b
b
xdx
x
1
1
1
1
x
x
xdx
x
Rationalize the numerator.
2
1+
1
xdx
x
2 2
1
1 1
xdx dx
x x
21u x
2 du x dx1
2du x dx
11 2
1sin
2x u du
2 2
1 x
1 1dx dx
x x
21u x
2 du x dx1
2du x dx
11 2
1sin
2x u du
1
1 2sin x u
1 2
1 0lim sin 1
b
bx x
1 2 1
1lim sin 1 sin 0 1b
b b
1
2
2
0 0
This integral converges because it approaches a solution.
Example 2:
1
0
dx
x
1
0lim ln
bbx
0lim ln1 lnb
b
0
1lim lnb b
This integral diverges.
(right hand limit)
We approach the limit from inside the interval.
1
0
1lim
bbdx
x
Example 3:
3
2031
dx
x
The function approacheswhen .
1x
2 233 3
01 1lim 1 lim 1
c
cc cx dx x dx
31 1
3 31 1
0
lim 3 1 lim 3 1c
c cc
x x
3
2031
dx
x
1 3
2 20 13 31 1
dx dx
x x
2 233 3
01 1lim 1 lim 1
c
cc cx dx x dx
31 1
3 31 1
0
lim 3 1 lim 3 1c
c cc
x x
11 1 133 3 3
1 1lim 3 1 3 1 lim 3 2 3 1c c
c c
0 0
33 3 2
Example 4:
1 P
dx
x
0P
1 Px dx
1lim
b P
bx dx
1
1
1lim
1
bP
bx
P
1 11lim
1 1
P P
b
b
P P
What happens here?
If then gets bigger and bigger as , therefore the integral diverges.
1P 1Pb
b
If then b has a negative exponent and ,therefore the integral converges.
1P 1 0Pb
(P is a constant.)
Day 1
Theorem 6 Comparison Test
Let and be continuous on , with 0 for all . Thenf g a f x g x x a
1. converges if converges.a af x dx g x dx
2. diverges if diverges.a ag x dx f x dx
1
xe dx
1lim
b x
be dx
1lim
bx
be
1lim b
be e
1 1lim
bb e e
0 1
e
Converges
Does converge?2
1
xe dx
Compare:
to for positive values of x.2
1xe
1xe
For2
2
1 11,
ex x
xxx e e
e
2
1xe
1xe
For2
2
1 11,
ex x
xxx e e
e
Since is always below , we say that it is
“bounded above” by .
2
1xe
1xe1
xe
Since converges to a finite number, must also converge!1xe
2
1xe
Example 7:
2
21
sin xdx
x
The maximum value of so:sin 1x
2
2 2
sin 10 on 1,
x
x x on
Since converges, converges.2
1
x
2
2
sin x
x
Example 7:
21
1
0.1dx
x
for positive values of x, so:2 0.1x x
Since diverges, diverges.1
x 2
1
0.1x
2
1 1
0.1 xx
on 1,
If functions grow at the same rate, then either they both converge or both diverge.
Does converge?21 1
dx
x
As the “1” in the denominator becomes
insignificant, so we compare to .
x
2
1
x
2
2
1
lim1
1
x
x
x
2
2
1limx
x
x
2lim
2x
x
x 1 Since converges,
converges.
2
1
x2
1
1 x
21 1
dx
x
Of course
21lim
1
b
b
dx
x
1
1lim tan
b
bx
1 1lim tan tan 1b
b
2 4
4
tany x
As , 2
y x
2
2
4
4
21 1
dx
x
21lim
1
b
b
dx
x
1
1lim tan
b
bx
1 1lim tan tan 1b
b
2 4
4
Of course 21
1 dx
x
2
1lim
b
bx dx
1
1lim
b
bx
1 1lim
1b b
1
0
Day 2
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