Integrating f(x)=xn

Understanding Integration of the Form f(x)=xn

The process of integration is often described as ‘anti-derivation’ or ‘anti-differentiation’.
The integral of a function is the antiderivative of that function.
The integral of the function f(x)=xn, where n ≠ -1, is F(x) = (xn+1/n+1) + C.
Notice the variable ‘C’ in the function. This is the constant of integration and represents an arbitrary constant whose derivative is zero.
The constant ‘C’ can be any constant value. It’s a critical part of any indefinite integral.
When integrating, we increase the power of ‘n’ by 1 and then divide by the new power.

Integration involves reversing the process of differentiation. If differentiating reduces the power by 1, integrating will increase it by 1.
Write down the function to be integrated, e.g., f(x)=x³.
Increase the exponent by one. In our example, this would turn x³ into x⁴.
Divide the result by the new exponent. Again following our example, we would have x⁴/4 as our result.
Finally, add the constant of integration ‘C’ to the result: x⁴/4 + C.

The integral of a function can represent a variety of different real-world quantities like areas, volumes, total value in economics, cumulative quantities, and more.
Integration is crucial in physics to find quantities like displacement, velocity, acceleration, and others.
In statistics, integration is used in the computation of expectation, variance and probability density functions.

Definite Integral: ∫ab f(x) dx = [F(b) - F(a)] where F(x) is the antiderivative of f(x).
Indefinite Integral of xn (n ≠ -1): ∫xn dx = (xn+1/n+1) + C.
Adding the integration constant: Always remember to include ‘C’, the constant of integration, when performing an indefinite integral.