Introduction to the integral of e^2x
In calculus, the integral is a fundamental concept that assigns numbers to functions to define displacement, area, volume, and all those functions that contain a combination of tiny elements. It is categorized into two parts, definite integral and indefinite integral. The process of integration calculates the integrals. This process is defined as finding an antiderivative of a function.
Integrals can handle almost all functions, such as trigonometric, algebraic, exponential, logarithmic, etc. This article will teach you what is integral to an exponential function e^2x. You will also understand how to compute the e 2x integral by using different integration techniques.
What is the integral of e to the power 2x?
The integral of e^(2x) is an antiderivative of the e^2x function which is equal to e^2x/2. The function e^2x is an exponential function and the integral of e is also known as its reverse derivative with respect to the variable involved. In simple words, we can say that finding the integration e2x with an integral calculator is a process of reversing the derivative of e^2x.
E^2x integration formula
The formula of integral of e^-2x contains integral sign, coefficient of integration and the function as sine. It is denoted by ∫(e^2x)dx. In mathematical form, the integral of e2x is:
$∫(e^{2x})dx = \frac{e^{2x}}{2} +c$
Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral.
How to evaluate integral of e^(2x) with respect to x?
The e2x integral is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate integral of e 2x by using:
- Integration by parts
- Definite integral
Each of these methods provides a different way to prove the e^2x integration. Moreover, these methods also help to compute the integral of e^-2x.
Integral of e to the 2x by using derivatives
The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. Therefore, we can use the derivative to calculate the integral of a function. Let’s discuss calculating the integration of e^2x by using derivatives.
Proof of integral of e^(2x) by using derivatives
Since we know that the integration is the reverse of the derivative. Therefore, we can calculate the integral of e^2x by using its derivative. For this, we have to look for some derivatives formulas or a formula that gives e^x as the derivative of any function.
In derivative, we know that,
$frac{d}{dx}(e^{2x}) =2e^{2x}$
It means that the derivative of e^2x gives us 2e^2x. Therefore, applying the integral on the both sides,
$\int\frac{d}{dx}(e^{2x})=2\int e^{2x}dx$
The integral sign at the left side will cancel out with the derivative of e^2x and the resulting expression will be,
$\int e^{2x}dx=\frac{e^{2x}}{2}+c$
Or,
$\int e^{2x}dx=\frac{e^{2x}}{2}+c$
Where c is a constant, known as the integration constant.
Integration of e^2x by using integration by parts
The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. The integration by parts is a method of solving the integral of two functions combined together. Let’s discuss calculating the integration e2x by using integration by parts.
Proof of integral of e to the 2x by using integration by parts
Since we know that the function e^2x can be written as the product of two functions. Therefore, we can integrate exponential e^2x by using integration by parts solver. For this, suppose that:
$u = e^{2x}$
Taking the derivatives, we get:
$\frac{du}{dx} = e^{2x}$
Plugging these values into the integration by parts formula, we get:
$∫e^{2x} dx =1∫e^{2x} dx - ∫[\frac{d}{dx}(1)∫e^{2x} dx]dx$
Simplifying,
$∫e^{2x} dx =\frac{e^{2x}}{2} - 0$
$∫e^{2x}dx =\frac{e^{2x}}{2}$
where C is the constant of integration. The calculations of the e^x integral is also same as above.
Integral of e 2x by using definite integral
The definite integral is a type of integral that calculates the area of a curve by using infinitesimal area elements between two points. The definite integral can be written as:
∫abf(x) dx = F(b) – F(a)
Let’s understand the verification of the integration of e^2x by using the definite integral.
Proof of integral of e^2x by using definite integral
To integrate exponential e^2x by using a definite integral, we can use the interval from 0 to 1. Let’s compute the integral of e^(2x) from 0 to 1. For this we can write the integral as:
$∫^1_0 e^{2x} dx = \frac{e^{2x}}{2}|^1_0$
Now, substituting the limit in the given function.
$∫^1_0 e^{2x} dx = \frac{e^2}{2} – \frac{e^0}{2}=\frac{e^2}{2}-\frac{1}{2}$
Since e^-2 = 0.135,
$∫^1_0 e^{2x} dx = \frac{0.135}{2}-\frac{1}{2}= -1.43$
Thus, the above is the calculation of the e2x integral. You can also use a definite integration finder for this.
FAQ’s
What is the anti derivative of e 2x?
The antiderivative of the integral of e 2x is equal to the ratio of e^2x and 2. Mathematically, it can be written as;
$\int e^{2x}=\frac{e^{2x}}{2}+c$
What is integral of e 3x?
The integral of e^3x is the integration of e^3x with respect to the variable x. It can be calculated by using integral formulas. Mathematically,
$\int e^{3x}=\frac{e^{3x}}{3}+c$