I am doing my IB math internal assessment for which I have to integrate the Gaussian distribution. I know I can just use a CDF on a calculator to find the probabilities I need but in order to get high level use of mathematics marks, I need to find the indefinite integral and then plug in the upper and lower bounds by hand. I tried to use wolfram alpha and some other online calculators but they either don't show steps or when they do, it's all unintuitive simplified mess of numbers. I also tried to follow youtube tutorials on how to integrate the normal distribution density function but there are so many different methods and it's hard to follow because I only know the basics of integration.

the function:

X ~ φ (112.99, 21.57)

if I plug that in the Gaussian function I get:

∫ (1/(21.57√(2π)))e^((-1/2)((x-112.99)/21.57)^2) dx

I have to find three probabilities, first one has an upper bound of 126.01 and lower bound of 112.99, the second one has an upper bound of 126.01 and lower bound of infinity, and the third one has a lower bound of 126.01 and an upper bound of infinity.

Is there an intuitive way to integrate this that I could follow?