# Integral of e^x

What is Integral of e^x. In mathematics, e is an empirical number. Its value is approximately 2.71828. It is also sometimes called ‘Euler’s number’. e is an important mathematical constant. This number is taken as the base of the natural logarithm. ,

## Definition of e

e is defined by the following two expressions-

{\displaystyle e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}}
{\displaystyle e=\sum _{k=0}^{\infty }{\frac {1}{k!}}={\frac {1}{0!}}+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+\cdots =1+1+{\frac {1}{2}}+{\frac {1}{6}}+{\frac {1}{24}}+\cdots }


#### Properties

e is an empirical irrational number .

##### Calculus

The exponential function ex is also important because it is the only function whose differential is also the same function. (so its anti-derivative is also the same)

{\displaystyle {\frac {d}{dx}}e^{x}=e^{x}}

{\displaystyle {\begin{aligned}e^{x}&=\int _{-\infty }^{x}e^{t}\,dt\\[8pt]&=\int _{-\infty }^{0}e^{t}\,dt+\int _{0}^{x}e^{t}\,dt\\[8pt]&=1+\int _{0}^{x} e^{t}\,dt.\end{aligned}}}

Euler’s formula

{\displaystyle e^{ix}=\cos(x)+i\,\mathrm {sin} (x),}

Substituting x = in this formula, we get Euler ‘s Identity-

{\displaystyle e^{i\pi }+1=0;}


#### Continuous fraction

{\displaystyle e-1=[1;0,1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,\ldots ]}

#### exponential decay

An amount undergoing exponential decay. The higher the value of the decay constant, the faster the value of the amount decreases. In the above graph the decays of the respective decay constants (λ) to 25, 5, 1, 1/5 and 1/25 on changing x from 0 to 5 are shown. A quantity will be studied as exponential decay if the quantity is decreasing directly proportional to its present value i.e. the rate of decrease of its value is directly proportional to its present value. Mathematically, the above statement can be expressed by the following differential equation, where N is the quantity and (lamda) is a positive number called the decay constant. The solution to the above equation is: Rate of change of exponential here N(t) ) is the quantity at time t and N0 .

#### empirical numbers

In mathematics, transcendental numbers are numbers that are not roots of any nonzero polynomial equation with rational coefficients. (Pi) and e are two prime empirical numbers. Proving that a given number is empirical is not easy. However, the empirical numbers are not rare. All real empirical numbers are irrational whereas not all irrational numbers are empirical. For example ‘the square root of 2’ is an irrational number but not an empirical number because it is the polynomial equation x2 − 2 .

## Integral of e^x

=~\int~e^x~dx
=~\int~e^x~dx~=~e^x~+C