Introduction to the integral of e^-x
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^-x. You will also understand how to compute the e^-x integral by using different integration techniques.
What is the integral of e^-x?
The integral of e^-x is an antiderivative of the e^-x function which is equal to e^-x. The function e^-x 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 integral of e to the -x is a process of reversing the derivative of e^-x.
Integral of e^-x formula
The formula of integral of e^-x contains integral sign, coefficient of integration and the function as sine. It is denoted by ∫(e^-x)dx. In mathematical form, the integral of e^-x is:
$$∫e^{-x}dx = -e^{-x} +c$$
Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral.
How to calculate the integral of e^-x?
The integral of e^-x is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate integral of e^-x by using:
- Integration by parts
- Definite integral
Integral of e 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 integral of e^-x by using derivatives.
Proof of integral of e^(-x) by using derivatives
Since we know that the integration is the reverse of the derivative. Therefore, we can calculate the integration of e^-x 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,
$$\frac{d}{dx}(e^{-x}) = -e^{-x}$$
It means that the derivative of e^-x gives us -e^-x. Therefore, applying the integral on the both sides,
$$\int \frac{d}{dx}(e^{-x})dx=\int e^{-x}dx$$
The integral sign at the left side will cancel out with the derivative of e^-x and the resulting expression will be,
$$\int e^{-x}dx=-e^{-x}+c$$
Where c is a constant, known as the integration constant. Also you can try our u-substitution calculator to integrate e^-x easily.
Integral of e^-x 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^(-x) integral by using integration by parts.
Proof of integral of e^-x by using integration by parts
Since we know that the function e^-x can be written as the product of two functions. Therefore, we can integrate it by using the integral by parts calculator. For this, suppose that:
$$u = e^{-x}$$
Taking the derivatives, we get:
$$\frac{du}{dx} = -e^{-x}$$
Plugging these values into the integration by parts formula, we get:
$$∫e^{-x} dx =1∫e^{-x} dx - ∫[\frac{d}{dx}(1)∫e^{-x} dx]dx$$
Simplifying,
$$∫e^{-x} dx =-e^{-x} - ∫(0)(-e^{-x})dx$$
$$∫e^{-x} dx =-e^{-x} - 0$$
$$∫e^{-x} dx =-e^{-x} + c$$
where C is the constant of integration.
Integral of e^-x 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 formula can be written as:
$$∫^b_a f(x) dx = F(b) – F(a)$$
Let’s understand the verification of the integration of e^-x by using the definite integral.
Proof of integral of e^-x by using definite integral
To compute the integral of e^-x by using a definite integral, we can use the interval from 0 to 1. Let’s compute the integral of e^(-x) from 0 to 1. For this we can write the integral as:
$$∫^1_0e^{-x} dx = -e^{-x}|^1_0 $$
Now, substituting the limit in the given function.
$$∫^1_0 e^{-x} dx = -e^{-1} + e^0 $$
Since e^0 is equal to 1 and e^1 is equal to 2.718, therefore,
$$∫^1_0 e^{-x} dx = -\frac{1}{2.718} + 1=0.63$$
FAQ’s
What is integration of e 2x?
The integral of e^2x is the same as the integration of e^-x. It is equal to the fraction of e^2x and the constant in its power i.e 2. Mathematically, it can be expressed as;
$$\int e^{2x}dx=\frac{e^{2x}}{2}+c$$
What is the formula for integral of Xe X?
The formula to integrate the product of two functions x and e^x is equal to the difference of xe^x and e^x. Mathematically, we can express the integral of xe^x as,
$$\int xe^x dx=xe^x-e^x+c$$