Power Rule of Integration is a fundamental law for finding the integrals of algebraic functions. It is used to find the integral of a variable raised to some constant power which may be positive, negative or zero except being -1.

In this article, we will discuss the power rule of integration, its mathematical representation, its applications, definite integration using the power rule, sample problems, practice problems and frequently asked questions related to the power rule of integration.

What is Integration?

Integration is the process of finding the integral of a function.

An integral is the anti-derivative of the function, i.e. if the derivative of F(x) is f(x), then the integral of f(x) would be F(x). Integral of a function is useful in finding the magnitude or expression of a physical quantity for any object given the expression of that quantity for a small region of that object subject to some technical constraints.

For example, if we have expression of area of a small strip on the object, we can derive expression for total area of that object by integrating that expression if the surface of the object is uniform.

Integration of f(x)

What is Power Rule of Integration?

Power Rule of integration states that integral of a variable x raised to a power n is the same variable x raised to power (n+1) divided by the power (n+1). This is a useful relation in determining the integral of monomial, binomial and polynomial algebraic functions.

Power Rule Formula

Mathematically, power rule of integration can be expressed as follows.

∫xⁿ dx = xⁿ⁺¹/(n+1) + C

Where, C is Constant of Integration.

Power rule of integration rule are applied for various functions that includes,

• Polynomial functions (like 11x³, x⁴, etc)
• Radical functions (like √x, ∛x, etc) as they can be written as exponents, i.e. (√x = x^1/2, ∛x = x^1/3)
• Some type of rational functions can be written in exponent form (like 1/x⁶, 1/x⁸, etc)

Read More about Power Rule of Derivatives.

Proof of Power Rule of Integration

Integration is the reverse process of differentiation and if the integral of a function F(x) is f(x), then differentiation of f(x) gives F(x). To prove power rule of integration, we must differentiate {(xⁿ⁺¹) / (n+1) + C} and if we get xⁿ then power rule is proved.

= d/dx{(xⁿ⁺¹) / (n+1) + C}

= d/dx{(xⁿ⁺¹) / (n+1)} + d/dx (C)

= 1/(n+1) d/dx (xⁿ⁺¹) + 0 (as d/dx(c) = 0)

= 1/(n+1)[(n + 1).x^n+1-1]

= xⁿ

Thus, d/dx ((xⁿ⁺¹) / (n+1) + C) = xⁿ

Hence,

∫xⁿ dx = (xⁿ⁺¹) / (n+1) + C

Thus, proved.

Now, let us discuss the applications of power rule of integration as follows.

Applications of Power Rule

Power Rule of Integration can be applied to find integral of various kinds of algebraic functions such as monomial functions, polynomial functions, radical functions and some type of rational functions. It is discussed in brief along with examples as follows:

Integration of Monomials Using Power Rule

Monomials are algebraic functions which involve single variable and a single term. For instance, 2x, x², 5x³, x⁵, etc. are examples of monomials. Let a monomial is expressed as k*xⁿ. Then, by power rule of integration, we can write,

∫k×xⁿ dx = k×xⁿ⁺¹/(n+1) + C

Example: ∫5x³ dx = ?

Solution:

∫5x³ dx = 5×x⁴/4 + C = 5x⁴/4 + C

Integration of Polynomials Using Power Rule

Polynomials are algebraic expression containing more than two terms involving addition or subtraction operations. The terms can be individually integrated and the signs of addition or subtraction remains the same before and after the integration.

Example:∫(3x² + 4x + 5)dx = ?

Solution:

∫(3x² + 4x + 5)dx = 3x³/3 + 4x²/2 + 5x + C = x³ + 2x² + 5x + C

Integration of Radicals Using Power Rule

Radicals are expressions in which a variable is raised to fractional power. Power Rule of Integration can be used in same way to find integral of Radical expressions too, i.e. adding one to the initial power and dividing by the new power to obtain integral of the given function.

Example: ∫x^1/3dx = ?

Solution:

∫x^1/3dx = x^1/3+1/(1/3+1) = x^4/3/(4/3) = 3x^4/3/4 + C

Integration of Rational Functions Using Power Rule

Rational functions which can be expressed as x^-n, where n is not equal to 1 can be integrated using power rule of integration, i.e. integral of x^-n would be x^-n+1/(1-n).

Example: ∫(1/x²)dx = ?

Solution:

∫(1/x²)dx = ∫x^-2dx = x^-1/-1 = -1/x + C

Integrating Negative Exponents Using Power Rule

Negative exponents are 1/a^m = a^-m. To integrate reciprocal functions we convert them to negative exponents and then integrate the same. This is explained as by the example added below as,

Example: ∫(1/x³)dx = ?

Solution:

∫1/x³ dx = ∫ x^-3 dx = (x^-3+1)/(-3+1) + C = (x^-2)/(-2) + C = -1/2x² + C

Definite Integrals Using Power Rule

Definite Integral of a function is integral of the function calculated within some specified limits called as upper limit and lower limit of integration. Power Rule of Integration can be used for finding definite integral of algebraic functions such as polynomial functions, radical functions and some type of rational functions. A general expression to find definite integral of a algebraic function with upper limit as b and a is presented as follows,

_a∫^bxⁿdx = [aⁿ⁺¹ – bⁿ⁺¹]/(n+1)