A proton is trapped in an infinite square well of width 15.0 nm. The "walls" of the well are at x = 0 and x = 15.0 nm. Assuming that the system is in the n = 2 state, calculate the probability of finding the proton at x = 12.0 ± 0.2 nm. Possibly integral from 11.8 to 12.2 = .027?

Hello!
Yes, this probability is an integral from 12 - 0.2 = 11.8 nm to 12 + 0.2 = 12.2 nm. There are two problems: what is the function to integrate and what is the value of the integral.
It is known that the function to integrate is the probability density, and it is the square of the wave function. Also it is known that the wave function for a particle in an infinite one-dimensional well with the walls x=0 and x=L is
Psi(x) = sqrt(2/L) sin((n pi x)/L),
where n is the state. So we need to integrate
p_d(x) = 2/Lsin^2((n pi x)/L) = 1/L (1-cos((2n pi x)/L)).
It is simple, and the probability is
int_(11.8)^(12.2) 1/L (1-cos((2n pi x)/L)) dx = (x/L - 1/(2n pi) sin((2n pi x)/L))|_(x=11.8)^(12.2)
Recall that n=2 and L=15nm and obtain
p = 0.4/15-1/(4pi)(sin(4pi*12.2/15)-sin(4pi*11.8/15)) approx 0.048.
(that said, your 0.027 is the first term, 0.4/15)