Integrating xn
In a nutshell
Integration is the inverse of differentiation. It can be used to work out a function given the gradient function, unique up to a constant of integration.
The rule for integration
Integration is the inverse of differentiation. The rule to differentiate xn is to multiply by the power, then subtract 1 from the power.
Therefore, the rule of integration is the inverse. First, add 1 to the power, then divide by the new power. Algebraically:
dxdy=axn⇒y=n+1axn+1
However, when differentiating a function, the constant term vanishes. There is no way to determine what this constant term could have been without more information.
y=x+1 y=x−1 y=x+5333218⇒differentiatedxdy=1⇒integratey=x⇒differentiatedxdy=1⇒integratey=x⇒differentiatedxdy=1⇒integratey=x
To acknowledge this loss of information, it is important to add a constant of integration after integrating, represented with a +C.
dxdy=axn f′(x)=axn⇒y=n+1axn+1+C⇒f(x)=n+1axn+1+Cn=−1n=−1
Note: This formula fails for n=−1. This is because if n=−1, then the denominator of the function is 0, which leads to the function being undefined. You are not expected to know how to integrate functions of the form x−1.
Examples
Find f(x) when:
i) f′(x)=4x
ii) f′(x)=x34
iii) f′(x)=16−5x
Part i):
Write the function in the form axn:
4x=4x1
Add 1 to the power and divide by this new power:
4x1→4x2→24x2→2x2
Add the constant of integration:
f(x)=2x2+C
Part ii):
Write the function in the form axn:
x34=4x−3
Add 1 to the power and divide by this new power:
4x−3→4x−2→−24x−2→−2x−2
Add the constant of integration:
f(x)=−2x−2+C
Part iii):
Integrate each term separately:
16=16x0 16x0→16x1→116x1→16x1 −5x=−5x1 −5x1→−5x2→2−5x2→−2.5x2
Add the terms together after integrating, remembering the constant of integration:
f(x)=16x−2.5x2+C