Just Find Its Antiderivative Right?

Calculus Level 4

Evaluate the Riemann integral

\[ \int_0^1 \frac{1}{ \sqrt{x} } \, dx . \]


Note: The Riemann integral of \(f\) equals \(S\) if the following condition holds:

\( \forall \epsilon > 0 \), \( \exists \delta \) such that for any tagged partition \( x_0, \ldots , x_n \) and \( t_0, \ldots , t_{n-1} \) whose mesh is less than \( \delta \), we have
\[ \left| \sum_{i=0}^{n-1} f(t_i) ( x_{i+1} - x_i) - S \right| < \epsilon. \]

Inspiration.

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