HP71B IBOUND fooled

[Albert Chan]

Code:

       / Inf
I1 =   |     sin(x)*sin(x^2) .dx
       / 0

We can solve above integral, using erf():

Let cis(x) = exp(i*x)

∫(cis(x^2))
= ∫(exp(-(w*x)^2)), where w = cis(-pi/4)
= sqrt(pi)/2 * erf(x*w)/w

∫(cis(x^2±x))
= ∫(cis((x±1/2)^2 - 1/4))
= cis(-1/4) * ∫(cis((x±1/2)^2))
= cis(-1/4) * sqrt(pi)/2 * erf((x±1/2)*w)/w

F(x) = ∫(sin(x)*sin(x^2)) = ∫(sinh(i*x)/i * sinh(i*x^2)/i)

= -1/4 * ∫((cis(x)-cis(-x)) * (cis(x^2)-cis(-x^2)))
= -1/4 * ∫((cis(x+x^2)+cis(-x-x^2)) - (cis(-x+x^2)+cis(x-x^2)))
= -1/2 * re(∫(cis(x^2+x) - cis(x^2-x)))
= -1/2 * re(cis(-1/4) * sqrt(pi)/2 * (erf((x+1/2)*w) - erf((x-1/2)*w))/w)

erf(∞) = 1 -> F(∞) = 0

erf() is odd -> F(0) = - re(cis(-1/4) * sqrt(pi)/2 * erf(w/2)/w)

Power Series Expansion of the Error Function
sqrt(pi/2)*erf(z) = z - z^3/(3*1!) + z^5/(5*2!) - z^7/(7*3!) + ...

sqrt(pi)/2 * erf(w/2)/w
= 1/2                - (-i)/(3*1!*2^3)     + (-1)/(5*2!*2^5)    - (+i)/(7*3!*2^7)
+ 1/(9*4!*2^9) - (-i)/(11*5!*2^11) + (-1)/(13*6!*2^13) - (+i)/(15*7!*2^15) + ...
= (1/2 - 1/320 + 1/110592 - 1/76677120 + ...)
+ (1/24 - 1/5376 + 1/2703360 - 1/2477260800 + ...) * i
≈ 0.496884029215 + 0.0414810242685*i

F(∞) - F(0)
= re(cis(-1/4) * sqrt(pi)/2 * erf(w/2)/w)
≈ cos(1/4) * 0.496884029215 + sin(1/4) * 0.0414810242685
≈ 0.491699677694