Download Improper Integrals: Convergence and Divergence and more Cheat Sheet Differential and Integral Calculus in PDF only on Docsity!
Improper Integrals
Let f : (a, b] โ R unbounded at x = a (x = a critical point) (f : [a, b) โ
R unbounded at x = b , x = b critical point) is a function integrable on
[a + , b] ([a, b โ ]) for all > o. We say that integral
โซ (^) b a f^ (x)dx^ converges
if limโ 0
โซ (^) b a+ f^ (x)dx^ (limโ^0
โซ (^) bโ a f^ (x)dx) exist.
Examples: Determine if the following integral is convergent or divergent.
0
โ^1 x
dx.
Critical point is x = 0 ( โ^1 x is not bounded at x = 0.
lim โ 0
0+
x
dx = lim โ 0
[2(
x|
1 )] = 2 lim โ 0 (
Thus integral
0
โ^1 x dx^ converges.
0
1 x^2 dx. Critical point is x = 0 ( (^) x^12 is not bounded at x = 0.
lim โ 0
x^2
dx = โ lim โ 0
x
|^1 ) = โ lim โ 0
Thus integral
0
1 x^2 dx^ diverges.
1
1 xโ 2 dx. x = 2 is critical point.
limโ 0
1
1 xโ 2 dx^ = limโ^0 (ln^ |x^ โ^2 ||
2 โ 1 ) = limโ^0 (ln(2^ โ^ x)|
2 โ 1 )
= limโ 0 (ln(2 โ (2 โ )) โ ln(2 โ 1)) = limโ 0 ln = โโ.
Thus integral
1
1 xโ 2 dx^ diverges.
1
2
Theorem 1Let f and g be continuous on (a, b) with 0 < f (x) โค g(x) for
all x โฅ a. Then
- If
โซ (^) b a g(x)dx^ converges, then^
โซ (^) b a f^ (x)dx^ converges
- If
โซ (^) b a f^ (x)dx^ diverges, then^
โซ (^) b a g(x)dx^ diverges
Theorem 2 Let positive functions f and g are continuous on (a, b] and
lim xโa
f (x)
g(x)
= L, 0 < L < +โ.
Then
โซ (^) b a f^ (x)dx^ and^
โซ (^) b a g(x)dx^ both converges or both diverges. Theorem 2โ Let positive functions f and g are continuous on [a, b) and
lim xโb
f (x)
g(x)
= L, 0 < L < +โ.
Then
โซ (^) b a f^ (x)dx^ and^
โซ (^) b a g(x)dx^ both converges or both diverges.
Integral โซ (^) b
a
(x โ a)p^
dx,
converges for p < 1 and diverges for p โฅ 1.
Integral โซ (^) b
a
(b โ x)p^
dx,
converges for p < 1 and diverges for p โฅ 1.
Examples: Determine if the following integral is convergent or divergent.
0
x+ x
โ x+
dx. Critical point is x = 0.
0 < f (x) =
x + 1
x
x + 2
x
x + 1 โ x + 2
