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Exponential Integral
In mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument. Definitions For real non-zero values of ''x'', the exponential integral Ei(''x'') is defined as : \operatorname(x) = -\int_^\infty \fract\,dt = \int_^x \fract\,dt. The Risch algorithm shows that Ei is not an elementary function. The definition above can be used for positive values of ''x'', but the integral has to be understood in terms of the Cauchy principal value due to the singularity of the integrand at zero. For complex values of the argument, the definition becomes ambiguous due to branch points at 0 and Instead of Ei, the following notation is used, :E_1(z) = \int_z^\infty \frac\, dt,\qquad, (z), 0. Properties Several properties of the exponential integral below, in certain cases, allow one to avoid its explicit evaluation through the definition ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Integrals Of Exponential Functions
The following is a list of integrals of exponential functions. For a complete list of integral functions, please see the list of integrals. Indefinite integral Indefinite integrals are antiderivative functions. A constant (the constant of integration) may be added to the right hand side of any of these formulas, but has been suppressed here in the interest of brevity. Integrals of polynomials * \int xe^\,dx = e^\left(\frac\right) \qquad \text c \neq 0; * \int x^2 e^\,dx = e^\left(\frac-\frac+\frac\right) * \begin \int x^n e^\,dx &= \frac x^n e^ - \frac\int x^ e^ \,dx \\ &= \left( \frac \right)^n \frac \\ &= e^\sum_^n (-1)^i\fracx^ \\ &= e^\sum_^n (-1)^\fracx^i \end * \int\frac\,dx = \ln, x, +\sum_^\infty\frac * \int\frac\,dx = \frac\left(-\frac+c\int\frac\,dx\right) \qquad\textn\neq 1\text Integrals involving only exponential functions * \int f'(x)e^\,dx = e^ * \int e^\,dx = \frac e^ * \int a^\,dx = \frac\qquad\texta > 0,\ a \ne 1 Integrals involving the er ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |