**Solar Scientific Calculators: Dealing with Integrals with Infinite Limits**

**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.

Source:

Mier-Jedrzejowic, W.A.C. Ph.D. __Extend Your 41__ London, UK 1985 ISBN 0-9510733-0-03

Eddie

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