Seriously tough definite integral?

Your answer is correct.

Consider a more general integral that depends on parameter a,

I(a) = ∫_-∞^∞ exp(-x^2) (erf(a*x))^2 dx.

Differentiate I(a) with respect to a and integrate the result by parts,

dI(a)/da = (2/√π) ∫_-∞^∞ 2 x exp(-(1+a^2) x^2) erf(a*x) d x =

- (2/√π) ( 1/(1+a^2)) ∫_-∞^∞ erf(a*x) d exp(-(1+a^2) x^2) =

(4/π) ( a /(1+a^2)) ∫_-∞^∞ exp(-(1+2a^2) x^2) dx,

or

dI(a)/da = (4/√π) a/[ (1+a^2) √(1+2a^2) ].

Since I(0) = 0, I(b) = ∫_0^b (dI(a)/da) da. Integration gives

I(a) = (4/√π) [ arctan( √(1 + 2a^2)) - π/4].

Using formulas for summation of arctan's we can rewrite this as

I(a) = (2/√π) arctan[a^2/√(1 + 2a^2)].

The answer follows after the substitution a = 1/√2.

I worked out a solution to this but it seems too clean, in my opinion however I will post my work and see if someone can spot anything wrong. I may leave out "trivial" steps for the sake of having a very linear solution, I am also trying to make this is as clean as possible to follow so I may do substitutions simply to clean up the look of the problem:

Geometrically:

A = ∫exp(-x²)*(erf(x/n)² dx

Integrated across the real number line, where n = sqrt(2)

A² = (∫exp(-x²)*(erf(x/n)² dx)²

A² = (∫exp(-x²)*(erf(x/n)² dx)²

Therefore

A² = (∫exp(-x²)*(erf(x/n)² dx)(∫exp(-y²)*(erf(y/n)² dy)

Where both x and y are integrated across the entire real number line.

Consider a transformation into polar coordinates:

r² = x² + y²

x = r cos(θ)

y = r sin(θ)

You can determine the element of integration by use of Jacobian, or simply know what it is from having a class that taught how to integrate on a polar graph.

dx dy -> r dr dθ

Keeping all of the above in mind we have:

A² = ∫∫exp(-(x²+y²))*(erf(x/n)²(erf(y/n)² dx dy

A² = ∫∫exp(-r²)*(erf((r cos(θ)/n)²(erf(r sin(θ)/n)² r dr dθ

Our limits of integration are transformed too: {(x,y) | (-∞, ∞), (-∞, ∞)} -> {(r, θ) | (0, ∞), (0, 2π)}

(This is not trivial, in my opinion and is referenced how to arrive at this conclusion)

The error function is:

erf(z) = m ∫ exp(-t²) dt, on {t | [0,z]} and where m = 2/sqrt(π)

So:

erf(z)² = (m ∫ exp(-t²) dt)² = M (∫ exp(-t²) dt) (∫exp(-s²) ds)

Where M = m², and {(s,t) | [0, z],[0, z]}

Thus for a polar transformation:

r'² = s² + t²

s = r' cos(φ)

t = r' sin(φ)

Here comes the tricky part (and where I think my problem may exist):

{(s,t) | [0,z], [0,z]} -> {(r', φ) | (0, z), (π/2, π/4)}

erf(z)² = M ∫∫ exp(-r'²) r' dr' dφ

erf(z)² = M ∫ dφ ∫ exp(-r'²) r' dr'

erf(z)² = M (-π/4) ∫ exp(-r'²) r' dr'

But M(-π/4) = m²(-π/4) = (2/sqrt(π))²(-π/4) = -1 (which is a cool result, or I made a mistake)

So our formula of interest is:

erf(z)² = - ∫ exp(-r'²) r' dr' = (exp(-z²) - 1)/2

Which follows that:

erf((r cos(θ)/n)² = (exp(- r² cos²(θ)/n²) - 1)/2

erf((r sin(θ)/n)² = (exp(- r² sin²(θ)/n²) - 1)/2

We know:

A² = ∫∫exp(-r²)*(erf((r cos(θ)/n)²(erf(r sin(θ)/n)² r dr dθ

Thus:

A² = ∫∫[exp(-r²)] [(exp(- r² cos²(θ)/n²) - 1)/2] [(exp(- r² sin²(θ)/n²) - 1)/2] r dr dθ

A² = (1/4)∫∫[exp(-r²)] [exp(- r² cos²(θ)/n²) - 1] [exp(- r² sin²(θ)/n²) - 1] r dr dθ

Which expands to:

A² = (1/4)∫∫(exp(-r²) - exp(-r² - (r² cos²θ)/n²) - exp(-r² - (r² sin²θ)/n²) + exp(-r² - (r² cos²θ)/n² - (r² sin²θ)/n²)) r dr dθ

From here out the goal is to separate these bad boys and call it a wrap.

A² = (1/4)∫∫(exp(-r²) - exp(-r² - (r² cos²θ)/n²) - exp(-r² - (r² sin²θ)/n²) + exp(-r² - (r² cos²θ)/n² - (r² sin²θ)/n²)) r dr dθ

This part is just a lot of algebra and assuming I am right up to this point, I think it is fair to just jump to an answer.

A² = (1/4)((1 + 2 n^2 - 2 sqrt(n² + n^4]) π)/(1 + n²)

Plugging n back in, and more ridiculous algebra leads to a final answer:

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A = [π (8/(4 + π) + π/(4 + π) - 4/(4 + π)^(1/2))]^(1/2)

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Note: I strongly feel like I made a mistake somewhere, I just can't spot were. But hopefully this gets the ball rolling towards an answer.