Problem with latest XKCD

[KeithB]

One of the numbers asked for in the latest XKCD is the integral from 0 to pi of x*sin^2 x.
https://xkcd.com/2966/

Explain xkcd says the answer is pi^2/4 or 2.4674...

https://www.explainxkcd.com/wiki/index.php/Main_Page

I tried with the HP71: Integral(0, pi, 1e-12, Ivar*sin(Ivar)*sin(Ivar)) = .007413..

The prime in cas mode gives an error

[KeithB]

never mind. I had degrees mode set.

[Dave Britten]

Okay, that "Game Theory" one definitely got a chuckle out of me.

[Albert Chan]

We can solve definite integral by folding:

\(\int_a^b f(x) \;dx =
\int_a^\frac{a+b}{2} (f(x) + f(a+b-x)) \;dx
\)

∫(x*sin(x)^2, x=0 .. pi) = ∫(x*sin(x)^2 + (pi-x)*sin(x)^2, x=0 .. pi/2) = pi * ∫(sin(x)^2, x = 0 .. pi/2)

We can use integral identities for RHS ... or we just fold again!

pi * ∫(sin(x)^2, x = 0 .. pi/2) = pi * ∫(sin(x)^2 + cos(x)^2, x = 0 .. pi/4) = pi * pi/4 = pi^2/4