Power Rule of Integration

The power rule of integration is one of the rules of integration and that is used to find the integral (in terms of a variable, say x) of powers of x. To apply the power rule of integration, the exponent of x can be any number (positive, 0, or negative) just other than -1.

Let us learn how to derive and apply the power rule of integration along with many more examples.

The power rule of integration is used to integrate the functions with exponents. For example, the integrals of x², x^1/2, x^-2, etc can be found by using this rule. i.e., the power rule of integration rule can be applied for:

• Polynomial functions (like x³, x², etc)
• Radical functions (like √x, ∛x, etc) as they can be written as exponents
• Some type of rational functions that can be written in the exponent form (like 1/x², 1/x³, etc)

The power rule says that: ∫ xⁿ dx = (xⁿ⁺¹) / (n+1) + C (where n ≠ -1).

To apply this rule, we simply add "1" to the exponent and we divide the result by the same exponent of the result. Finally, add C to the final result (the integration constant). Here are some examples of this rule:

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

We know that integration is the reverse process of differentiation and if the integral of a function F(x) is f(x), then differentiating f(x) gives F(x) back. So, to prove the power rule of integration, we just integrate the result (xⁿ⁺¹) / (n+1) + C and see whether we get xⁿ back.

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 the derivative of a constant is 0)
= 1/(n+1) [ (n + 1) x^n+1-1] (by power rule of derivatives)
= xⁿ ( ∵ (n+1) has got canceled)

Thus, d/dx ((xⁿ⁺¹) / (n+1) + C) = xⁿ and hence ∫ xⁿ dx = (xⁿ⁺¹) / (n+1) + C. Hence, proved.

The power rule is meant for integrating exponents and polynomial involves exponents of a variable. Hence, the power rule is applied to integrate polynomial functions. In this process, we may have to apply the properties of integrals (like ∫ c f(x) dx = c ∫ f(x) dx). For example, f(x) = 2x² - 3x is a polynomial function and we can apply the power rule and properties of integrals as shown below to integrate this.

∫ (2x² - 3x) dx = ∫ 2x² dx - ∫ 3x dx (∵ ∫ (f(x) + g(x)) dx = ∫f(x) dx + ∫ g(x) dx)
= 2 ∫ x² dx - 3 ∫ x¹ dx (∵ ∫ c f(x) dx = c ∫ f(x) dx)
= 2 (x³/3) - 3 (x²/2) + C (by power rule of integration)
= (2x³)/3 - (3x²)/2 + C

We have a property of negative exponents that says 1/a^m = a^-m. This is the main property that is used to integrate the reciprocal functions by converting them as negative exponents. For example, ∫ 1/x² dx = ∫ x^-2 dx and by integrating this using power rule, we get ∫ x^-2 dx = (x^-2+1)/(-2+1) + C = (x^-1)/(-1) + C = -1/x + C. Here are some more examples:

• ∫ 3x^-4 dx = 3 ∫ x^-4 dx = 3 (x^-3/(-3)) + C = -1/x³ + C
• ∫ (5x^-2)/3 dx = 3 ∫ x^-2 dx = (5/3) (x^-1/(-1)) + C = -5/(3x) + C

Note: We cannot integrate ∫ (1/x) dx using the power rule by writing it as ∫ x^-1 dx. Because, if we apply the power rule for this, we get x⁰/0 + C. But x⁰/0 is not defined. So the power rule of integration cannot be applied just when the exponent is -1. Note that ∫ (1/x) dx = ln x + C.

A radical is of the form ⁿ√x and this can be written as x^1/n. This representation helps to convert a radical into exponent form. Thus, it is possible to integrate radicals using the power rule of integration. Here are some examples.

• ∫ √x dx = ∫ x^1/2 dx = (x^3/2) / (3/2) + C = (2 x^3/2) / 3 + C
• ∫ ∛x dx = ∫ x^1/3 dx = (x^4/3) / (4/3) + C = (3 x^4/3) / 4 + C

So far, we have understood that the power rule of integration is useful whenever we see an exponent and we can find the integrals like ∫ x dx, ∫ x^-3 dx, ∫ x^1/2 dx, etc using this rule. But this rule is used to find the integrals of non-zero constants and the integral of zero as well. Let us learn more about this.

• The integral of 1 is x + C. This is because we know that 1 = x⁰ and
  ∫ 1 dx = ∫ x⁰ dx = (x⁰⁺¹)/(0+1) + C = x¹ + C = x + C
• The integral of any constant with respect to x is the product of that constant and x. But add C at the end. For example,
  ∫ 2 dx = 2 ∫ 1 dx = 2 x + C
• The integral of 0 is C. This is because
  ∫ 0 dx = 0 ∫ 1 dx = 0(x) + C = C

Important Notes on Power Rule of Integration:

• The power rule of integration is used to integrate the terms that are of the form "variable raised to exponent".
• By the power rule, the integral of xⁿ is (xⁿ⁺¹) / (n+1) + C.
• The power rule of integration can't be applied when n = -1.
• We can integrate polynomials, negative exponents, and radicals using the power rule.

☛ Related Topics:

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What is the Formula of Power Rule of Integration?

The formula for power rule of integration says ∫ xⁿ dx = (xⁿ⁺¹) / (n+1) + C, where

• 'n' is any real number other than -1 (i.e., 'n' can be a positive integer, a negative integer, a fraction, or a zero).
• C is the integration constant.

When to Use the Power Rule of Integration?

The power rule of integration is one of the integration rules that is used to integrate a term that has an exponent in it. Here, the exponent can be any number other than -1.

What is the Example of the Power Rule of Integration?

The power rule in integration is ∫ xⁿ dx = (xⁿ⁺¹) / (n+1) + C. For applying this rule, simply add 1 to the given exponent and divide by the same resultant exponent. Add a C at the end. For example, ∫ x⁵ dx = (x⁶) / 6 + C.

What is the Difference Between the Power Rule of Differentiation and Integration?

• The power rule of differentiation is used to differentiate the exponents. It says d/dx (xⁿ) = n x^n-1.
• The power rule of integration is used to integrate the exponents. It says ∫ xⁿ dx = (xⁿ⁺¹) / (n+1) + C.