Hello everyone how are you today. I want to give a tutorial for you
Integrals with Infinite Limits
Today's post deals with integrals with infinite limits in the forms:
∫( f(x) dx, x = a to x = ∞)
∫( f(x) dx, x = -∞ to x = ∞)
∫( f(x) dx, x = -∞ to x = a)
One method to deal with these integrals, as suggested by W.A.C. Mier-Jedrzejowicz Ph. D. (see the source), is to use the substitution
x = tan θ
Then:
dx = dθ/cos^2 θ
and θ = atan x.
Also, as x approaches π/2, tan x approaches +∞.
And, as x approaches -π/2, tan x approaches -∞.
With the substations, let's test four integrals on four solar-powered scientific calculators:
1. Casio fx-991EX Classwiz
2. Sharp EL-W516T
3. Texas Instruments TI-36X Pro
4. Casio fx-115ES Plus
Set the calculator to radians mode.
Example 1: ∫(1/x^2 dx, x = 1 to x = ∞) = 1
∫(1/x^2 dx, x = 1 to x = ∞)
with the substitutions x = tan θ and dx = dθ/(cos^2 θ):
∫( 1/tan^2 θ * dθ/cos^2 θ, θ = atan 1 to θ = π/2)
∫( 1/sin^2 θ * dθ, θ = atan 1 to θ = π/2)
We can evaulate the integral straight away. Here are the results:
1. Casio fx-991EX Classwiz
Time: 1.37 seconds
Answer: 1
2. Sharp EL-W516T
Time: 38 seconds
Answer: 1
3. Texas Instruments TI-36X Pro
Time: 4.5 seconds
Answer: 1
4. Casio fx-115ES Plus
Time: 4.2 seconds
Answer: 1
A promising start.
Example 2: ∫(e^(-0.5*x^2), x = 0 to x = ∞) ≈ 1.25331413732
∫(e^(-0.5*x^2), x = 0 to x = ∞)
with the substituions, this becomes:
∫(e^(-0.5 * tan^2 θ)/cos^2 θ dθ, θ = atan 0 to θ = π/2)
atan 0 = 0
But look at the denominator, we have cos^2 θ. Since cos^2 π/2 = 0, there will be a problem. Let's use an approximation of π/2 of 1.5708.
∫(e^(-0.5 * tan^2 θ)/cos^2 θ dθ, θ = 0 to θ = 1.5708)
Here are the results:
Integrals with Infinite Limits
Today's post deals with integrals with infinite limits in the forms:
∫( f(x) dx, x = a to x = ∞)
∫( f(x) dx, x = -∞ to x = ∞)
∫( f(x) dx, x = -∞ to x = a)
One method to deal with these integrals, as suggested by W.A.C. Mier-Jedrzejowicz Ph. D. (see the source), is to use the substitution
x = tan θ
Then:
dx = dθ/cos^2 θ
and θ = atan x.
Also, as x approaches π/2, tan x approaches +∞.
And, as x approaches -π/2, tan x approaches -∞.
With the substations, let's test four integrals on four solar-powered scientific calculators:
1. Casio fx-991EX Classwiz
2. Sharp EL-W516T
3. Texas Instruments TI-36X Pro
4. Casio fx-115ES Plus
Set the calculator to radians mode.
Example 1: ∫(1/x^2 dx, x = 1 to x = ∞) = 1
∫(1/x^2 dx, x = 1 to x = ∞)
with the substitutions x = tan θ and dx = dθ/(cos^2 θ):
∫( 1/tan^2 θ * dθ/cos^2 θ, θ = atan 1 to θ = π/2)
∫( 1/sin^2 θ * dθ, θ = atan 1 to θ = π/2)
We can evaulate the integral straight away. Here are the results:
1. Casio fx-991EX Classwiz
Time: 1.37 seconds
Answer: 1
2. Sharp EL-W516T
Time: 38 seconds
Answer: 1
3. Texas Instruments TI-36X Pro
Time: 4.5 seconds
Answer: 1
4. Casio fx-115ES Plus
Time: 4.2 seconds
Answer: 1
A promising start.
Example 2: ∫(e^(-0.5*x^2), x = 0 to x = ∞) ≈ 1.25331413732
∫(e^(-0.5*x^2), x = 0 to x = ∞)
with the substituions, this becomes:
∫(e^(-0.5 * tan^2 θ)/cos^2 θ dθ, θ = atan 0 to θ = π/2)
atan 0 = 0
But look at the denominator, we have cos^2 θ. Since cos^2 π/2 = 0, there will be a problem. Let's use an approximation of π/2 of 1.5708.
∫(e^(-0.5 * tan^2 θ)/cos^2 θ dθ, θ = 0 to θ = 1.5708)
Here are the results:
1. Casio fx-991EX Classwiz
Time: 15.4 seconds
Answer: 1.253314137
2. Sharp EL-W516T
Time: 1 minute, 8 seconds
Answer: errors out
3. Texas Instruments TI-36X Pro
Time: 36 seconds
Answer: 1.253314138
4. Casio fx-115ES Plus
Time: 1 minute, 6.8 seconds
Answer: 1.253314137
Example 3: ∫(x^2*e^-x dx, x = 0 to x = ∞) = 2
∫(x^2*e^-x dx, x = 0 to x = ∞)
with the substitutions and simplification, we get:
∫( (sin^2 θ * e^(-tan θ))/cos^4 θ dθ, θ = 0 to θ = π/2)
Like the last situation, there is a potential problem with the denominator. Let's see if we can use an approximation of π/2, this time using 1.57 in hopes to cut the calculation time down.
∫( (sin^2 θ * e^(-tan θ))/cos^4 θ dθ, θ = 0 to θ = 1.57)
Here are the results:
1. Casio fx-991EX Classwiz
Time: 27 seconds
Answer: 2
2. Sharp EL-W516T
Time: 1 minute, 34 seconds
Answer: 1.999999999
3. Texas Instruments TI-36X Pro
Time: 1 minute, 9 seconds
Answer: 2
4. Casio fx-115ES Plus
Time: 1 minute, 6.8 seconds
Answer: 1.253314137
Example 4: ∫( e^-x/x^2 dx, x = 1 to x = ∞) ≈ 0.148495506776
∫( e^-x/x^2 dx, x = 1 to x = ∞)
with the substitutions and simplification, we get:
∫( e^(-tan θ)/sin^2 θ dθ, θ = 0 to θ = π/2)
I'm going to use the 1.57 approximation again and set the integral as:
∫( e^(-tan θ)/sin^2 θ dθ, θ = 0 to θ = 1.57)
Here are the results:
1. Casio fx-991EX Classwiz
Time: errors out immediately
Answer: N/A
2. Sharp EL-W516T
Time: 1 minute, 6 seconds
Answer: error
3. Texas Instruments TI-36X Pro
Time: 7 seconds
Answer: 0.148495519
4. Casio fx-115ES Plus
Time: errors out after 1 second
Answer: N/A
Some Observations
1. Not all calculations of improper integrals will be successful.
2. Out of the four calculators tested, from the four calculations: the Casio fx-991ES is the fastest, but I found the most successful with the Texas Instruments TI-36X Pro.
3. Be ready to spend a little for calculations by using this method.
thanks you don't forget to comments and shared this article
Solar Scientific Calculators: Dealing with Integrals with Infinite Limits # New 2019
Reviewed by dicky
on
11:50
Rating:
No comments: