Integral of xe^x

The integral of xe^x is equal to xe^x - e^x + C, where C is the constant of integration. Mathematically, we can write the formula for the integral of xe^x (OR xe^x) as ∫xe^x dx = xe^x - e^x + C. This formula can be derived using the integration by parts method of integration. We can choose the first and second function using the sequence ILATE for applying the integration by parts formula and hence, determine the integral of xe^x.

Further, in this article, we will evaluate the integral of xe^x and find its formula using the integration by parts method. We will also determine the definite integral of xe^x with different limits and solve a few examples to evaluate the integral of functions including function xe^x for a better understanding of the concept.

The integral of xe^x (also, written as integral of xe^x) is equal to xe^x - e^x + C. The integral of a function is nothing but the reverse process of differentiation. Therefore, the integral of xe^x is also called the antiderivative of xe^x. It can be calculated using one of the important methods of integration, known as the integration by parts method. The integral of a function gives the area under the curve of the function. So, we can say that the integral of xe^x gives the area under the curve of the function f(x) = xe^x. Let us go through the formula of the integral of xe^x in the next section:

The formula for the integration of xe^x is given by, ∫xe^x dx = xe^x - e^x + C (OR) ∫xe^x dx = e^x(x - 1) + C, where C is the integration constant, ∫ is the symbol of integration and dx shows the integral of xe^x is with respect to x. The image below shows the simplified formula of integral of xe^x and it can be evaluated using the ILATE method (also known as integration by parts method).

Now, we know that the formula for the integral of xe^x is given by, ∫xe^x dx = e^x(x - 1) + C. We will derive this formula in this section using the integration by parts method (Product rule of integration). We will use the formula ∫f(x) g(x) dx = f(x) ∫g(x) dx - ∫[df/dx × ∫g(x) dx] dx. Here, we choose f(x) and g(x) according to ILATE - Inverse function, Logarithmic function, Algebraic function, Trigonometric function, and Exponential function. So, the first function f(x) = x (Because x is an algebraic function) and g(x) = e^x (Because e^x is an exponential function). We will also use the following formulas:

• Integral of e^x: ∫e^x dx = e^x + C
• Derivative of x: d(x)/dx = 1

Using the formulas, we have

∫xe^x dx = x ∫e^x dx - ∫[dx/dx × ∫e^x dx] dx

= xe^x - ∫(1 × e^x) dx

= xe^x - ∫e^x dx

= xe^x - e^x + C

= e^x (x - 1) + C

Hence, we have derived the formula for the integral of xe^x using the method of integration by parts.

In this section, we will evaluate the definite integral of xe^x with limits ranging from 0 to 1. We will substitute these limits into the formula of the integral of xe^x to find its definite integral. We know that ∫xe^x dx = e^x(x - 1) + C, where C is the constant of integration. So, we have

₀ ∫¹ xe^x dx = [ e^x(x - 1) + C ]₀¹

= (e¹(1 - 1) + C) - (e⁰(0 - 1) + C)

= (e × 0 + C) - (1 × -1 + C)

= C + 1 - C

= 1

Hence, the definite integral of xe^x from 0 to 1 is equal to 1.

Important Notes on Integral of xe^x

• The integral of xe^x is equal to e^x(x - 1) + C, where C is the integration constant.
• We can calculate the integral of xe^x using the method of integration by parts.
• The definite integral of xe^x from 0 to 1 is equal to 1.

☛ Related Topics:

• Derivative of exponential function
• Derivative of 2^x
• Differentiation and Integration

What is Integral of xe^x in Calculus?

The integral of xe^x is equal to xe^x - e^x + C, where C is the constant of integration. The integral of xe^x gives the area under the curve of the function f(x) = xe^x. We can evaluate this integral using the integration by parts formula.

What is the Formula of Integral of xe^x?

The formula for the integration of xe^x is given by, ∫xe^x dx = xe^x - e^x + C (OR) ∫xe^x dx = e^x(x - 1) + C, where C is the integration constant. We can calculate this integral by using the ILATE sequence of function in integration by parts method.

How to Find the Integral of xe^x?

We can find the integral of xe^x using one of the most commonly used and important methods of integration known as the integration by parts. We can use the formula of integration by parts given by, ∫f(x) g(x) dx = f(x) ∫g(x) dx - ∫[df/dx × ∫g(x) dx] dx OR ∫udv = uv - ∫vdu.

What is the Definite Integral of xe^x From 0 to 1?

The definite integral of xe^x from 0 to 1 is equal to 1. This value can be determined by substituting the limits and 0 and 1 into the formula ∫xe^x dx = e^x(x - 1) + C and find its value.

What is the Integral of x^e^x^2?

The integral of x^e^x^2 is given by, ∫xe^x2 dx = (1/2) e^x2 + C. This formula of integral of x^e^(x^2) can be calculated using the substitution method of integration.

How Do You Find Integral of xe^x2?

We can find the integral of xe^x2 using the substitution method of integration. We can assume x² to be equal to some variable u (say) and change the variable of integration to simplify the integral problem.

Is the Antiderivative of xe^x the Same as the Integral of xe^x?

Antidifferentiation is the reverse process of differentiation, which is also known as integration. Therefore, the antiderivative of xe^x is the same as the integral of xe^x. The antiderivative rules can be applied to find the integral of a function.