Mastering the Power Rule of Integration for Evaluating Integrals of x Raised to the Power of n

∫ xⁿ dx

To evaluate the integral of x raised to the power of n, we can use the power rule of integration

To evaluate the integral of x raised to the power of n, we can use the power rule of integration. The power rule states that when integrating x raised to the power of a constant, the result is (x^(n+1))/(n+1), where n is any real number except -1.

So, when integrating x raised to the power of n, we add 1 to the exponent and then divide by the new exponent:

∫ xⁿ dx = (x^(n+1))/(n+1) + C

Here, C represents the constant of integration.

Let’s go through a couple of examples to illustrate:

1. ∫ x² dx
Using the power rule, we add 1 to the exponent (2 + 1 = 3) and then divide by the new exponent:
= (x^(2+1))/(2+1) + C
= (x^3)/3 + C

2. ∫ x⁴ dx
Again, using the power rule, we add 1 to the exponent (4 + 1 = 5) and then divide by the new exponent:
= (x^(4+1))/(4+1) + C
= (x^5)/5 + C

In both examples, we have evaluated the integral of x raised to the power of n using the power rule and obtained a result in the form of (x^(n+1))/(n+1) + C, where C is the constant of integration.

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