two methods, two answers

[nomadreid]

There are two methods to take d/dx [ ∫_t=0^x exp(-t^2) dt].

First method: using the relationship of integral and antiderivative, one gets
(exp(-t^2) , from t = 0 to x, so exp(-x^2) - 1.

Second method: the integral is (1/2)sqrt(pi)*erf(t) from 0 to x, which is (1/2)sqrt(pi)*erf(x), and the derivative of this is exp(-x^2).

So, which answer is correct, and what is wrong with the other method?

 

[galactus]

By the Fundamental Theorem of Calculus, the second one is correct.

The first one is incorrect. \(\displaystyle e^{-t^{2}}\) does not have an elementary antiderivative so

\(\displaystyle \left e^{-t^{2}}\right|_{0}^{x}\neq e^{-x^{2}}-1\)

This is why mathematicians have came up with erf. So as to define \(\displaystyle \int e^{-x^{2}}dx\)


\(\displaystyle \displaystyle \frac{d}{dx}\int_{0}^{x}e^{-t^{2}}dt=e^{-x^{2}}\)

 

[nomadreid]

Thanks, galactus.