Integral of e^x

The integral of ex is a function used in calculus because its derivative and integral are both the same. The function ex is written mathematically as:

              ∫ e^xdx = e^x + C

Where: 

∫ is the integral symbol

e is Euler’s number, which is approximately 2.718.

ex is the function being integrated. It represents exponential growth.

dx is when we integrate with respect to x, and x is the variable that changes.

C is the constant of integration, which represents an infinite number of possible solutions.

The integral of ex formula gives the area under the curve, which is ex + C.

There are many ways to find the integral of ex. In this section, we will talk about the common methods we use to find the integral of ex:

• By differentiation
   
• Using Series Expansion
   
• Definite integral of e^x

We know that differentiation and integration are the reverse of each other. To find the integral of e^x, we identify the function whose derivate is e^x. If we use the formula for differentiation: d/dx (e^x) = e^x

We can directly say that the integral of ex is ex itself. We can prove this by the fundamental theorem of calculus.

Integrate both sides by dx,

d/dx (e^x) dx = e^x dx

Integrate both sides of the equation,

∫ d/dx (e^x) dx = ∫ e^xdx

By fundamental theorem, differentiation and integration are the inverse of each other, so d/dx and ∫ ddx cancel each other out. 

We get,

 e^xdx = e^x + C

Hence we prove the integral of ex formula.

All standard functions have series expansions. For e^x the series expansion is, 

Step 1: Integrate both sides

e^x = 1 + x + x²/2! + x³/3! + …. Take integral on both sides, we will get
∫ e^xdx =  [1 + x +x2/2! + x3/3! + ….]dx
 
 

Step 2: Apply the Power Rule of Integration

By the power rule of integration,

= x + x²/2 + x³/3(2!) + x⁴/4(3!) +  ….

= x + x2/2! + x3/3! + x4/4! + …

 

Step 3: Recognizing the Series

So, adding and subtracting 1 inside the sum,

∫ e^xdx = 1 + x +x2/2! + x3/3! + x4/4! + ... - 1 

We know that 1 + x + x2/2! + x3/3! + x4/4! + ... = ex.
 

Here, we can replace the constant -1 by the integration constant C. Thus,

∫ e^xdx = e^x + C

Hence, Proved.

To know the definite integral of ex we can ignore the integration constant C and substitute the bounds in ex. 
Integral of ex from 0 to 1
01 exdx = [ex]01
= e1 - e0
= e - 1
Thus the integral of ex from 0 to 1 is e - 1.

Integral of ex from 0 to ∞
0∞ exdx = [ex]0∞
= e∞ - e0
= ∞ - 1
= ∞
Thus the integral of ex from 0 to ∞ diverges.

Learning integrals can be quite confusing. To master integration of e^x here are some tips and tricks that students can use when solving problems involving ex.

• Remember the rule that the integral of ex is the easiest to remember: 

• ∫ e^xdx = e^x + C. The function e^x is unique because its derivative and integral are identical, unlike most other functions.

•  When there is an exponent that has a coefficient like e^3x, just divide by the coefficient itself. 

• When solving for a definite integral from a to b just remove C and subtract the value at limits. 

• ∫ab e^xdx = e^b - e^a.

• Integral: Represented by the symbol. It is the process of finding the antiderivative of a function. It finds the area under a curve or the accumulation of a quantity.

• Euler’s Number (e): It is a special constant used in exponential and logarithmic functions. It has a constant value of approximately 2.718.

• Exponential function: It is a mathematical function expressed as ex, where e is the base and x is the exponent.