Integral of XE^X

Integral of xe^x along with its formula and proof with examples. Also learn how to calculate integration of xe^x with step by step examples.

Alan Walker-

Published on 2023-04-20

Introduction to the integral of xe^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 xe^x. You will also understand how to compute xe^x integral by using different integration techniques.

What is the integral of xe^x?

The integral of xe^x is an antiderivative of the xe^x function which is equal to xe^x – e^x. It is also known as the reverse derivative of xe^x function which is an exponential funciton. 

The sine function is the ratio of the opposite side to the hypotenuse of a triangle which is written as:

Sin = opposite side / hypotenuse 

Integral of xe^x formula

The formula of the xe^x integral contains the integral sign, coefficient of integration, and the function as sine. It is denoted by ∫(xe^x)dx. In mathematical form, the integral of x e^x is:

$∫(xe^x)dx = -xe^x - e^x +c$

 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 xe^x?

The integral of xe^x is its antiderivative, which can be calculated using different integration techniques. In this article, we will discuss how to calculate the integration of xe^x by using:

  1. Integration by parts
  2. Definite integral

Integral of xe^(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. The formula to calculate the integral by using the integral by parts calculator is;

$\int f(x)g(x)dx=f(x)\int g(x)dx - \int \left(\frac{d}{dx}(f(x).\int g(x)dx\right)dx$

Let’s discuss calculating the xe^(x) integral by using integration by parts.

Proof of integral of xe^x by using integration by parts

Since we know that the function xe^x can be written as the product of two functions. Therefore, we can integrate it by using integration by parts. For this, suppose that:

$u = xe^x$

Taking the derivatives, we get:

$\frac{du}{dx} = xe^x$

Plugging these values into the integration by parts formula, we get:

$∫xe^x dx =x∫e^x dx - ∫[\frac{d}{dx}(x)∫e^x dx]dx$

Since the integral of e^x is e^x. Now, simplify, 

$∫xe^x dx =xe^x - ∫(1)e^xdx$

$∫xe^x dx =xe^x - ∫e^xdx$

$∫xe^x dx =xe^x - e^x + C$

where C is the constant of integration. Our integration calculator also helps to calculate the step-by-step integration of a function. 

Xe^x integral 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 xe^(x) by using the definite integral.

Proof of integration of xe^x by using definite integral

To compute the integral of $x e^x$ by using a definite integral, we can use the interval from 0 to 1. Let’s compute the xe^(x) integral from 0 to 1. For this, we can write the integral as:

$∫^1_0 xe^x dx = xe^x|^1_0$

Now, substitute the limit in the given function.

$∫^1_0 xe^x dx = (1)e^1 – e^1 - (0(e^0)-e^0)$

Since cos 0 is equal to 1 and cos π is equal to -1, therefore, 

$∫^1_0 xe^x dx = -(-1)= 1$

Try the definite and improper integral calculator to find the integral of $xe^x$ in easy steps. 


What is the integral of ex dx?

The integral of an exponential function e^x is equal to itself, i.e. e^x. It is expressed mathematically as;

$\int e^x dx=e^x+c$

What is the antiderivative of xe^x?

The antiderivative or integration of xe^x is equal to xe^x - e^x + C. It can be calculated by using interation by parts and the definite integral. Mathematically, it is expressed as:

$\int xe^xdx=xe^x - e^x+c$

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