Integral of x

Integration of x refers to the process of determining the value of the integral of x by using different integration methods. Integration can be defined as the process of finding the antiderivative of a function. There are two main methods to find the integration of x. These are the power rule and integration by parts.

In this article, we will evaluate the definite and indefinite integral of x, find the integration of x using various methods as well as study various associated examples.

Integration of x refers to the process of finding the antiderivative of x. In other words, integration of x is the same as finding the antiderivative of x. The general form of integration is given as ∫f(x) dx. Here, " ∫" depicts the sign of integration. f(x) denotes the function that is being integrated and dx is used to show the variable with respect to which the function is being integrated. Thus the integration of x is represented as follows:

∫x dx

The formula for integration of x is given as ∫x dx = \(\frac{x^{2}}{2}\) + C. where C denotes the constant of integration. Integrals are used to represent a family of curves which implies that the value of the integral will not be unique. This is the reason why a constant of integration is required. It is used to show that the integrals of a family of curves differ by a constant number.

The easiest method to find the integral of x is by using the power rule. This rule states that the integral of x raised to a power n will be equal to x raised to the power n + 1 divided by n + 1. The formula for the power rule of integration is given as follows:

\(\int x^{n}dx = \frac{x^{n+1}}{n+1} + C\)

Using this formula the integration of x can be performed as follows:

As the exponent of x is 1 thus, n = 1. Substituting these values in the integration power rule we get,

\(\int x^{1}dx = \frac{x^{1+1}}{1+1} + C\)

Thus, the formula for the integral of x is ∫x dx = \(\frac{x^{2}}{2}\) + C.

When two functions are in product form then to calculate the value of the integral, integration by parts is used. Integration by parts is also known as the product rule. Suppose there are two function f(x) and g(x) then the formula for integration by parts is given as follows:

∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫ (f′(x) ∫g(x) dx) dx + C

Here f'(x) denotes the differentiation of the first function and g'(x) shows the differentiation of the second function.

By taking f(x) = x and g(x) = 1, the integration of x using the product rule can be determined.

∫x. 1 dx = x ∫1 dx − ∫ (1 ∫1 dx) dx + C. (Using the rules of differentiation, f'(x) = 1)

∫x. 1 dx = x . x - ∫ (1 . x) dx + C

∫x. 1 dx = x² - ∫x. 1 dx + C

2 ∫x. 1 dx = x² + C

∫x dx = \(\frac{x^{2}}{2}\) + C.

Thus, this is the formula for the integration of x.

An integral that does not have any specified limits is known as an indefinite integral. Thus, ∫x dx is an indefinite integral. However, if the integral needs to be evaluated between two points then definite integrals are used. The general for of a definite integral is given as follows:

\(\int_{a}^{b}f(x) dx = F(b)-F(a)\)

Here, a is the lower limit and b is the upper limit. As the integration is performed between specified limits the constant of integration gets removed.

Suppose the integral of x needs to be determined between points 2 and 4 then the steps to find the values of the definite integral of x are as follows:

• Express the definite integral of x as \(\int_{2}^{4}xdx\)
• Perform the integration using the power rule. This gives \(\left [ \frac{x^{2}}{2} \right ]_{2}^{4}\). As it is a definite integral the constant of integration is not used.
• Substitute the lower limit in the function from the previous step. Thus, (2)² / 2 = 2.
• Substitute the upper limit in the function. This gives (4)² / 2 = 8.
• Subtract the value obtained in step 3 from the value obtained in step 4. This gives 8 - 2 = 6.

The generalized formula for the definite integration of x is \(\int_{a}^{b}xdx = \frac{b^{2}}{2} - \frac{a^{2}}{2}\)

Related Articles:

• Integration Formulas
• Integration Formula for Substitution
• Integration by Partial Fractions

Important Notes on Integration of x

• On performing the integration of x the value obtained is (x² / 2) + C, where C is the constant of integration.
• The value of the integral of x can be computed using the power rule, \(\int x^{n}dx = \frac{x^{n+1}}{n+1} + C\), with n = 1.
• Using the product rule the value of the integral of x can be determined by taking f(x) = x and g(x) = 1.
• The definite integral of x is used when the integration has to be performed between two specified limits. It is given as \(\int_{a}^{b}xdx = \frac{b^{2}}{2} - \frac{a^{2}}{2}\).

What is Integration of x in Calculus?

The integration of x in calculus refers to the process of computing the value of the antiderivative of x. The product rule and the power rule are used to determine the value of the integral of x.

What is the Formula for Integration of x?

The formula for the integration of x is ∫x dx = \(\frac{x^{2}}{2}\) + C where C is the constant of integration.

How to Find the Integral of x?

The integral of x can be computed by using the power rule and the product rule. Using n = 1 in the power rule formula, \(\int x^{n}dx = \frac{x^{n+1}}{n+1} + C\), the value of the integral can be determined. Furthermore, by taking f(x) = x and g(x) = 1 the product rule, given by ∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫ (f′(x) ∫g(x) dx) dx + C ,can be used to find the integral value.

What is the Definite Integral of x?

When the integration of x has to be evaluated between two specified points the definite integral of x is used. This is given as \(\int_{a}^{b}xdx = \frac{b^{2}}{2} - \frac{a^{2}}{2}\) where a is the lower limit and b is the upper limit.

What is the Integral of x Cube?

To find the integral of x cube, the power rule is used. This is given as \(\int x^{3}dx = \frac{x^{3+1}}{3+1} + C\) = x⁴ / 4 + C

What is the Derivative and Integral of x?

Using the differentiation power rule the derivative of x will be 1. The integral of x is (x² / 2) + C

What Does C Denote in the Integral of x?

C denotes the constant of integration in the integral of x. It is used to show that the various integrals of a family of curves differ from each other by a constant.