Properties of Integrals

Properties of integrals define the rules for working across integral problems. The properties of integrals can be broadly classified into two types based on the type of integrals. They are the properties of indefinite integrals, and the properties of definite integrals.

Let us learn more about the different properties of integrals, and their examples.

The properties of integrals are helpful in solving integral problems. Integration involving functions and algebraic expressions. These problems can be solved with a thorough knowledge of the properties of integrals. The properties of integrals can be broadly classified as the following two types based on the types of integrals.

• Properties of indefinite integrals
• Properties of definite integrals.

Let us understand the properties of each of these integrals.

The following are the five important properties of indefinite integrals.

• The process of integration and differentiation are reverse to each other. \(\frac{d}{dx}.\int f(x).dx = f(x) \) OR \(\int f'(x).dx = f(x) + C\), where C is an arbitrary constant.
• Two indefinite integrals with the same derivative, if they are equal, then their function representing the family of curves are equivalent.\(\frac{d}{dx} \int f(x).dx = \frac{d}{dx} \int g(x).dx \), then f(x) is equivalent to g(x).
• The integration of the sum of two functions, is equal to the sum of the integration of the individual functions.\(\int [f(x) + g(x)].dx = \int f(x).dx + \int g(x).dx\)
• For a real number k, the integration of the product of k and the function is equal to the product of constant k and the integral of the function. \(\int k.f(x).dx = k \int f(x).dx\)
• The integration of the summation of the constant and the function is equal to the summation of the integrals of the product of the constant and the function. \(\int [k_1.f_1(x) + k_2.f_2(x) + .....k_nf_n(x)].dx = k_1 \int f_1(x).dx + k_2\int f_2(x).dx + ....k_n\int f_n(x).dx \).

The properties of definite integrals are helpful to integrate the given function and apply the lower and the upper limit to find the value of the integral. The definite integral formulas help for finding the integral of a function multiplied by a constant, for the sum of the functions, and for even and odd functions. Let us check below, some of the important properties of definite integrals.

• \(\int ^b_a f(x) .dx = \int^b _a f(t).dt \)
• \(\int ^b_a f(x).dx = - \int^a _b f(x).dx \)
• \(\int ^b_a cf(x).dx = c \int^b _a f(x).dx \)
• \(\int ^b_a( f(x) \pm g(x)).dx = \int^b _a f(x).dx \pm \int^b_ag(x).dx\)
• \(\int ^b_a f(x) .dx = \int^c _a f(x).dx + \int^b_cf(x).dx\)
• \(\int ^b_a f(x) .dx = \int^b _a f(a + b - x).dx \)
• \(\int ^a_0 f(x) .dx = \int^a _0 f(a - x).dx \) (This is a formula derived from the above formula.)
• \(\int^{2a}_0f(x).dx = 2\int^a_0f(x).dx\) if f(2a - x) = f(x)
• \(\int^{2a}_0f(x).dx = 0\) if f(2a - x) = -f(x).
• \(\int^a_{-a}f(x).dx = 2\int^a_0f(x).dx\) if f(x) is an even function.
• \(\int^a_{-a}f(x).dx = 0\) if f(x) is an odd function, and f(-x) = -f(x).

Related Topics

The following topics help in a better understanding of the properties of integrals.

• Integration by Parts
• Calculus Formulas
• Indefinite Integral
• Definite Integral
• Integration by Substitution
• Integration by Partial Fractions

What Are the Properties of Integrals?

The properties of integrals are helpful to solve the numerous problems of integrals. The properties of integrals can be classified as properties of indefinite integrals, and properties of definite integrals. A few of the important properties of integrals are as follows.

• \(\int ^b_a f(x).dx = - \int^a _b f(x).dx \)
• \(\int ^b_a f(x) .dx = \int^c _a f(x).dx + \int^b_cf(x).dx\)
• \(\int ^b_a f(x) .dx = \int^b _a f(a + b - x).dx \)
• \(\int^{2a}_0f(x).dx = 2\int^a_0f(x).dx\) if f(2a - x) = f(x)
• \(\int^a_{-a}f(x).dx = 2\int^a_0f(x).dx\) if f(x) is an even function
• The integration of the sum of two functions, is equal to the sum of the integration of the individual functions.\(\int [f(x) + g(x)].dx = \int f(x).dx + \int g(x).dx\)
• For a real number k, the integration of the product of k and the function is equal to the product of constant k and the integral of the function. \(\int k.f(x).dx = k \int f(x).dx\)

What Are the Properties of Definite Integrals?

The important properties of definite integrals, which are helpful to work across definite integrals, are as follows.

• \(\int ^b_a f(x).dx = - \int^a _b f(x).dx \)
• \(\int ^b_a f(x) .dx = \int^c _a f(x).dx + \int^b_cf(x).dx\)
• \(\int ^b_a f(x) .dx = \int^b _a f(a + b - x).dx \)
• \(\int^{2a}_0f(x).dx = 2\int^a_0f(x).dx\) if f(2a - x) = f(x)
• \(\int^a_{-a}f(x).dx = 2\int^a_0f(x).dx\) if f(x) is an even function, and f(-x) = f(x)


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What Are the Properties of Indefinite Integrals?

The three important properties of indefinite integrals are as follows.

• The indefinite integrals with the same derivative, if they are equal, then their function representing the family of curves are equivalent.\(\frac{d}{dx} \int f(x).dx = \frac{d}{dx} \int g(x).dx \), then f(x) is equivalent to g(x).
• The integration of the sum of two functions, is equal to the sum of the integration of the individual functions.\(\int [f(x) + g(x)].dx = \int f(x).dx + \int g(x).dx\)
• For a real number k, the integration of the product of k and the function is equal to the product of constant k and the integral of the function. \(\int k.f(x).dx = k \int f(x).dx\)

What Are the Rules of Integrals?

The rules of integrals refer to all the properties of integrals, and integral formulas. A few of the important formulas of integrals are as follows.

• \(\int ^b_a f(x).dx = - \int^a _b f(x).dx \)
• \(\int^{2a}_0f(x).dx = 2\int^a_0f(x).dx\) if f(2a - x) = f(x)
• \(\int^a_{-a}f(x).dx = 2\int^a_0f(x).dx\) if f(x) is an even function, and f(-x) = f(x).
• The integration of the sum of two functions, is equal to the sum of the integration of the individual functions.\(\int [f(x) + g(x)].dx = \int f(x).dx + \int g(x).dx\)
• For a real number k, the integration of the product of k and the function is equal to the product of constant k and the integral of the function. \(\int k.f(x).dx = k \int f(x).dx\)

What Is the Symbol of Integrals?

The symbol of integrals is almost similar to the elongated form of the alphabet 'S'. The symbol of integration is \(\int \).