∫a^b e^-u^2 d u =∫a^0 e^-u^2 d u+∫0^b e^-u^2 d u=∫0^b e^-u^2 d u-∫0^a e^-u^2 d u =(√(π))/

step 1 so the question says use the laplace transform and table 1 .5 .1 to solve the integral equation

∫a^b e^-u^2 d u =∫a^0 e^-u^2 d u+∫0^b e^-u^2 d u=∫0^b...

$$\begin{aligned}\int_{a}^{b} e^{-u^{2}} d u &=\int_{a}^{0} e^{-u^{2}} d u+\int_{0}^{b} e^{-u^{2}} d u=\int_{0}^{b} e^{-u^{2}} d u-\int_{0}^{a} e^{-u^{2}} d u \\&=\frac{\sqrt{\pi}}{2} \operatorname{erf}(b)-\frac{\sqrt{\pi}}{2} \operatorname{erf}(a)=\frac{\sqrt{\pi}}{2}[\operatorname{erf}(b)-\operatorname{erf}(a)] \end{aligned}$$

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Key Concepts

Error Function

The error function, often denoted as erf(x), is a special mathematical function defined by the integral (2/??) ??? e^(-t²) dt. It is used to represent the antiderivative of the Gaussian function and plays a crucial role in areas such as probability theory and statistics, where it relates to the cumulative distribution of the normal distribution.

Splitting of Integrals

Splitting the integral involves breaking a definite integral over an interval into the sum or difference of integrals over subintervals. This technique, which takes advantage of the additivity property of integration, allows for the convenient application of functions like the error function when integrating across symmetric or specific segments of the domain.

Gaussian Integral

The Gaussian integral refers to the integration of the function e^(-u^2) over a given interval. This integral is fundamental in probability, statistics, and physics, particularly because the antiderivative of e^(-u^2) cannot be expressed in terms of elementary functions. Instead, it is linked to special functions such as the error function.

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