RPN: Certain Integrals to Positive Infinity
Introduction
Today’s RPN session deals with improper integrals where the upper limit is positive infinity (∞).
∫( f(x) dx, a, ∞) → lim ∫( f(x) dx, a, t) as t → ∞
Some properties that will be use:
lim 1/t^n as t → ∞ approaches 0
lim e^(-t) as t → ∞ approaches 0
lim p(t) ÷ q(t) as t → ∞ approaches 0 where p(t) and q(t) are polynomials and
degree p(t) < degree q(t)
lim p(t) ÷ q(t) as t → ∞ approaches p_n ÷ q_n where p(t) and q(t) are polynomials and
degree p(t) = degree q(t). p_n and q_n are the leading coefficients of p(t) and q(t), respectively.
All integrals presented will have closed formulas because they have very friendly anti-derivatives.
∫( 1 ÷ (x^n) dx, a, ∞)
∫( 1 ÷ (x^n) dx, a, ∞) = 1 ÷ ((n - 1) * a^(n - 1))
(Abbreviated) Derivation:
∫( 1 ÷ (x^n) dx, a, ∞)
= lim (1 ÷ x^(n - 1) * -1 ÷ (n – 1) as x → ∞) - (1 ÷ a^(n – 1) * -1 ÷ (n – 1))
= ( 0 * -1 ÷ (n – 1) ) + 1 ÷ ((n – 1) * a^(n – 1))
= 1 ÷ ((n – 1) * a^(n – 1))
∫( 1 ÷ (x^n) dx, a, ∞) = 1 ÷ ((n - 1) * a^(n - 1))
HP 15C Code:
[42, 21, 11]: LBL A
[ 1]: 1
[ 30]: -
[ 43, 36]: LSTx
[ 20]: ×
[ 15]: 1/x
[ 43, 32]: RTN
Stack:
Y: a
X: n
Examples:
a = 2.75, n = 2: 4/11 ≈ 0.36364
a = 4.9, n = 3: ≈ 0.02082
∫( 1 ÷ ((x - r)*(x - s)) dx, a, ∞)
∫( 1 ÷ ((x - r)*(x - s)) dx, a, ∞)
= 1 ÷ (r - s) * ln( abs((a - s) ÷ (a – r)) )
For best results, a > max(s, r)
(Abbreviated) Derivation:
∫( 1 ÷ ((x - r)*(x - s)) dx, a, ∞)
Simply by partial fractions:
1 ÷ ((x - r)*(x – s)) = 1 ÷ ((r – s) * (x – r)) – 1 ÷ ((r – s) * (x – s))
Anti-derivative:
∫( 1 ÷ ((x - r)*(x - s)) dx)
= 1 ÷ (r – s) * ( ln(abs(x – r)) – ln(abs(x – s)) )
= 1 ÷ (r – s) * ln ( abs((x – r) ÷ (x – s)) )
Limit as x → ∞:
1 ÷ (r – s) * ln ( abs((x – r) ÷ (x – s)) )
= 1 ÷ (r – s) * ln ( abs((1 – r ÷ x) ÷ (1 – s ÷ x)) )
= 1 ÷ (r – s) * ln ( abs(1) )
= 1 ÷ (r – s) * ln(1)
= 0
When x = a
1 ÷ (r – s) * ln ( abs((a – r) ÷ (a – s)) )
= 1 ÷ (r – s) * ln ( abs(1 ÷ [(a – r) * (a -s)]) )
= 1 ÷ (r – s) * ln ( 1 ÷ [abs((a – s) ÷ (a – r))] )
= -1 ÷ (r – s) * ln ( abs((a – s) ÷ (a – r)) )
Then:
∫( 1 ÷ ((x - r)*(x - s)) dx, a, ∞) = 0 - (-1 ÷ (r – s) * ln ( abs((a – s) ÷ (a – r)) ))
= 1 ÷ (r - s) * ln ( abs((a - s) ÷ (a – r)) )
Code:
[42, 21, 12]: LBL B
[ 44, 1]: STO 1
[ 33]: R↓
[ 44, 2]: STO 2
[ 33]: R↓
[ 44, 3]: STO 3
[45, 30, 1]: RCL- 1
[ 45, 3]: RCL 3
[45, 30, 2]: RCL- 2
[ 10]: ÷
[ 43, 16]: ABS
[ 43, 12]: LN
[ 45, 2]: RCL 2
[45, 30, 1]: RCL- 1
[ 15]: 1/x
[ 20]: ×
[ 43, 32]: RTN
Stack:
Z: a
Y: r
X: s
Examples:
a = 7.25, b = -3, s = 6: (ln 41 - ln 5) ÷ 9 ≈ 0.23379
a = 11, b = 4, s = 9: (ln 7 - ln 2) ÷ 5 ≈ 0.25055
∫( 1 ÷ e^x dx, a, ∞)
∫( 1 ÷ e^x dx, a, ∞) = 1 ÷ e^a
Code:
[42, 21, 13]: LBL C
[ 12]: e^x
[ 15]: 1/x
[ 43, 32]: RTN
Stack:
X: a
Examples:
a = 4: e^(-4) ≈ 0.01832
a = 6: e^(-6) ≈ 0.00248
∫( x ÷ e^x dx, a, ∞)
∫( x ÷ e^x dx, a, ∞) = (a + 1) ÷ e^a
Code:
[42, 21, 14]: LBL D
[ 12]: e^x
[ 43, 36]: LSTx
[ 1]: 1
[ 40]: +
[ 34]: x<>y
[ 10]: ÷
[ 43, 32]: RTN
Stack:
X: a
Examples:
a = 4: 5 ÷ e^(-4) ≈ 0.09158
a = 6: 7 ÷ e^(-6) ≈ 0.01735
Eddie
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