∫ x^(n) dx
The integral of x to the power of n, denoted as ∫ x^n dx, can be calculated using the power rule of integration
The integral of x to the power of n, denoted as ∫ x^n dx, can be calculated using the power rule of integration. The power rule states that when integrating a function of the form x^n, where n is any real number except -1, you can increase the exponent by 1 and divide the result by the new exponent.
So, applying the power rule to the integral of x^n, we get:
∫ x^n dx = (x^(n+1))/(n+1) + C
where C is the constant of integration.
For example, let’s say you want to integrate x^3. Using the power rule, we have:
∫ x^3 dx = (x^(3+1))/(3+1) + C
= (x^4)/4 + C
Similarly, if you want to integrate x^(-2), we have:
∫ x^(-2) dx = (x^(-2+1))/(-2+1) + C
= x^(-1)/(-1) + C
= -1/x + C
It’s important to note that the power rule does not apply when n = -1. In that case, you would use the natural logarithm function to integrate x^(-1).
More Answers:
The Power Rule of Integration: Finding the Integral ∫ k dx with Step-by-Step InstructionsMastering Integration Techniques: A Comprehensive Guide to ∫ kf(x) dx
Integrating the Sum or Difference of Two Functions: A Step-by-Step Guide