1. p-Integrals In Practice
4 Confirm Theorem 3.1 Analytically (use anti-derivatives):
1
Show ⌠
dx 1
4.1  x= 1=2
⌡ 1-
0 2
1
Show ⌠
dx 1 3
4.2 = =
3 1 2
⌡ x 1-3
0
a:pintprac.doc 1

2. p-Integrals In Practice
5 Confirm Theorem 3.3 Analytically (use anti-derivatives):
∞
Show ⌠ 2 =
dx 1
5.1 =1
⌡ x 2-1
1
∞
Show ⌠ 3 =
dx 1 1
5.2 =
⌡ x 3-1 2
1
a:pintprac.doc 2

3. p-Integrals In Practice
6 Confirm Corollary 3.2 Numerically (use convergence/divergence tables):
1
Show ⌠
dx
6.1 Diverges.
⌡ x2
0
1
Show ⌠ 3 Diverges.
dx
6.2
⌡x
0
a:pintprac.doc 3

4. p-Integrals In Practice
7 Confirm Corollary 3.2 Numerically (use convergence/divergence tables):
∞
Show ⌠
dx
7.1  Diverges.
⌡ x
1
∞
Show ⌠
dx
7.2 Diverges.
3
⌡ x
1
a:pintprac.doc 4

5. p-Integrals In Practice
8 For each of the following p-Integrals, apply the appropriate Theorem or Corollary
to demonstrate convergence or divergence. Using a method of your choice
(analytical or numerical) confirm your results.
1
8.1 ⌠ dx
3
⌡ x2
0
∞
8.2 ⌠ dx
3
⌡ x2
1
1
⌠ dx
8.3  x3
⌡
0
∞
⌠ dx
8.4  x3
⌡
1
a:pintprac.doc 5