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 = x1/2, ∛x = x1/3)
- 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).xn+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: ∫x1/3dx = ?
Solution:
∫x1/3dx = x1/3+1/(1/3+1) = x4/3/(4/3) = 3x4/3/4 + 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/am = 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)
Sample Problems on Power Rule of Integration
Problem 1: Find the integral of function, f(x) = 3x2.
Solution:
We have,
f(x) = 3x2
Using Power Rule of Integration, we get,
∫3x2dx
= 3×x3/3 + C
= x3 + C
Thus, we obtain ∫3x2dx = x3 + C
Problem 2: Evaluate the definite integral 1∫34x2dx.
Solution:
Let,
I = 1∫34x2dx
⇒ I = [4x3/3]13
⇒ I = (1/3)×[4×33 – 4×(1)3]
⇒ I = 104/3
Thus, value of given integral comes out to be 104/3.
Problem 3: Find the integral of the function f(x) = x2/3+√x.
Solution:
We have,
f(x) = x2/3 + √x = x2/3 + x1/2
⇒ ∫f(x)dx = x2/3+1/(2/3+1) + x1/2+1/(1/2+1)
⇒ ∫f(x)dx = x5/3/(5/3) + x3/2/(3/2)
⇒ ∫f(x)dx = 3x5/3/5 + 2x3/2/3
Problem 4: Find the integral of the function f(x) = 1/x4.
Solution:
We have,
f(x) = 1/x4
⇒ ∫f(x)dx = ∫(1/x4)dx = ∫(x-4)dx
⇒ ∫f(x)dx = x-4+1/(-4+1) = x-3/(-3) = (-1/3)x-3
⇒ ∫f(x)dx = -1/3x3+ C
Practice Problems on Power Rule of Integration
P1: Find the integral of the function f(x) = x2 + 1/x2.
P2: Find the integral of the curve represented by the function, f(x) = √x.
P3: If p(x) = 3x2+4x+5, find an expression for ∫p(x)dx.
P4: Integrate the function f(x) = x1/3 within limits as x = 0 and x = 1.
P5: Evaluate ∫(1/√x)dx.
FAQs on the Power Rule of Integration
What is Power Rule of Integration?
Power Rule of Integration states that integral of an expression of the form xn is given by xn+1/(n+1). It is mathematically expressed as, ∫xndx = xn+1/(n+1) + C
Are there any exceptions to Power Rule of Integration?
Power Rule of Integration is to find integral of a expression in which a variable is raised to some constant. The constant can be any rational number except (-1).
Why does Power Rule of Integration fail for exponent being -1?
When exponent is -1, then the function is 1/x. Integral of 1/x does not converge for all x, so it can’t be calculated by power rule of integration. It is calculated by antiderivative principle as ln x, where x belongs to the set of positive real numbers.
Can Power Rule of Integration be used to evaluate definite integrals as well?
Yes, definite integrals can also be evaluated using power rule of integration. One can find the integral using the power rule and then substitute limits to evaluate the integral.
Can Power Rule be applied directly to find integral of composite functions?
No, the Power Rule cannot be directly applied to composite functions involving algebraic operations on trigonometric functions or exponential functions. There are different rules and techniques, such as trigonometric identities or integration by substitution to find integral of those functions.
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