Integral of 0

The integral of 0 is equal to an arbitrary constant as the derivative of a constant function is always equal to zero. Before going to calculate the integral of zero, let us recall about integration. Since integration is the reverse process of differentiation, we can use the differentiation itself to do the integration, if the function that we need to integrate is very simple like 1, 0, x, etc. i.e., for knowing what is the result of the integration of 0, we have to think by differentiating what function would result in 0.

Let us learn more about finding the integral of 0 in different ways along with a few examples for a better understanding of the concept.

The integral of 0 is C, where C is a constant. This is mathematically written as ∫ 0 dx = C. Here,

• The integrand is 0.
• dx lets us know that the integration of zero is with respect to x.
• C is the integration constant.

Let us see how to derive the integration of 0 in two ways:

• using differentiation
• using power rule of integration

For finding the integral of 0 using the process of differentiation, think by differentiating what expression would give 0. i.e., think to fill the question mark in the following equation:

d/dx ( ? ) = 0

We know that the derivative of any constant is 0. So, we have d/dx (C) = 0, where C is a constant. Taking the integral on both sides, we have

∫ d/dx (C) dx = ∫ 0 dx

By the fundamental theorem of calculus, the integral (along with dx) and derivative get canceled. So we get

C = ∫ 0 dx

Hence, we have derived the formula of integration of zero.

Verification of Integral of 0

To verify the integral of 0, we just differentiate the result and see whether we get 0 back. Since the ∫ 0 dx = C, let us find the derivative of C. Then, d/dx (C) = 0 (by derivative rules). Therefore, the integral of 0 is C and is verified.

Let us consider the integral ∫ 0 dx. We can write this integral as 0 ∫1 dx. We know that the integral of 1 is x + C using the power rule of integration. So

∫ 0 dx = 0 ∫ 1 dx

= 0 (x + C)

= 0

(OR)

∫ 0 dx = 0 ∫1dx

= 0 ∫x⁰ dx

= 0 [x⁰⁺¹/(0+1)] + C

= 0 + C

= C

Since the integration constant is added to every indefinite integral's value,

∫ 0 dx = 0 + C = C.

Hence, the integral of 0 formula is proved.

The definite integral of 0 is the integral with two (lower and upper) limits. Let us consider a definite integral with the lower limit a and upper limit b. i.e., ∫ₐ^b 0 dx. Since ∫ 0 dx = C, the value of the definite integral is obtained by substituting the upper and lower limit in the result (C) and subtracting the results. Then

∫ₐ^b 0 dx = [C]ₐ^b = C - C = 0

So the definite integral of 0 is always equal to 0 irrespective of the limits.

Important Notes on Integral of 0:

• The integral of 0 is C. i.e., ∫ 0 dx = C.
• The definite integration of 0 from a to b gives 0. i.e., ∫ₐ ^b 0 dx = 0.

Related Topics:

• Derivative Formulas
• Derivative Calculator
• Differentiation of Trigonometric Functions
• Inverse Trig Derivatives

What is the Value of Integration of 0?

The integral of 0 is C. It is written as ∫ 0 dx = C, where C is the integration constant.

How to Do the Integration of Zero?

To find the integral of 0, just see the derivative formulas and see by differentiating what term would give 0. We have d/dx (C) = 0, where C is a constant. Hence the ∫ 0 dx = C.

Is the Antiderivative of 0 Equal to 0 Itself?

No, the antiderivative of 0 is equal to C. We know that the antiderivative is also known as integral and hence the integral of 0 is C which is written as ∫ 0 dx = C. So, the antiderivative of 0 is not equal to 0.

What is the Value of the Integral of 0 With Bounds?

We know that ∫ 0 dx = C. If we take 'a' to be its lower bound and 'b' to be its upper bound, then ∫ₐ^b 0 dx = C - C = 0. So the value of integral of zero with any bounds is 0.

How to Find the Definite Integral of 0?

We have ∫ 0 dx = C. Now, we consider the definite integral, ∫ₐ^b 0 dx and to evaluate this, we substitute x = b and then x = a in the result (C) and find the difference. Then ∫ₐ^b 0 dx = (C) - (C) = 0.