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 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 of e^(-2x) is a process of reversing the derivative of e^-2x.
Integral of e^(-2x) formula
The formula to integrate e^-2x contains the integral sign, coefficient of integration, and the function as exponential. It is denoted by ∫(e^-2x)dx. In mathematical form, the integral of e^-2x 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 the integral.
How to calculate the integral of e^-2x?
The integral e^-2x is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate the integral of e^-2x by using:
- Integration by parts
- Definite integral
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 both sides,
$\int \frac{d}{dx}(e^{-2x})dx=-2\int e^{-2x}dx$
The integral sign at the left side will cancel out with the derivative of e^-2x and the resulting expression will be,
$-2\int e^{-2x}dx=e^{-2x}+c$
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 e^(-2x) integral by using integration by parts.
Proof of integration of e^(-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 it by using the integration by parts calculator. 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.
Integration of e power -2x by using definite integral
The definite integral is a type of integration that calculates the area of a curve by using infinitesimal area elements between two points. The definite integral formula calculator can be written as:
$∫^b_a f(x) dx = F(b) – F(a)$
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 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, substitute 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,
$∫^1_0 e^{-2x} dx = -0.135/2+½= -0.03375$
If the integral is not bounded between two points, use our indefinite integration solver to evaluate such integrals easily.
FAQ's
What is the integral of ex?
The integral of e^x is equal to e^x. It is because e^x is an exponential function whose integral is equal to the ratio between the function and its power. Mathematically, it can be written as;
$\int e^x dx=e^x+c$