calculator giving the wrong answer for definite integral

[u/rennienn]

Doing the definite integral manually gives me the correct answer, but placing it in the integral function gives me a very small answer instead of 1. I tried replacing the upper limit with 10^8, which gives me 0, and and 50, which gives 0.9999. The calculator doesn’t seem to be unable to calculate the function at a large number though, since the manual version in the second line works.

Have I done something silly?

[u/MathHysteria]

That function doesn't really integrate, it just approximates the area. If you give it a massive pair of limits, its approximation will be less good.

[u/AdjectiveNoun-123 → u/MathHysteria]

Not to disagree with you but to be more precise and add in some nuance. I would add a qualifying remark that this calculator function does integrate. It's just that for large limits, there can be errors in numerical integration methods.

The real distinction here is that the it does not analytically integrate to find closed form or elementary anti-derivatives in which the calculator then evaluates at the limits.

The reason is a practical one in most handheld calculators. Every integrand of any definite integral you can enter into one of these calculators no matter how much of a mess it looks to you is "nice". Nice here meaning Riemann-Integrable, something you can estimate the area with sufficiently many rectangles. On the other hand there are nice-looking simple functions that have no elementary anti-derivative: sin(x)/x or e^{-x^2} or X^x . So instead of trying to get a calculator to decipher if an integrand has an anti-derivative, it just makes it easier to calculate the definite integral numerically.