This just means, integrate with respect to . Remember, add one to the power and divide by the new power.
The appears because when you differentiate a constant term, the answer is zero, so as we are performing 'anti-differentiation', we presume there may have been a constant term, which reduced to zero when differentiated. This is called the constant of integration.
- In other words you add one to the power, divide by the new power and add the constant of integration.
Integration is the inverse process to differentiation. So instead of multiplying by the index and reducing the index by one, we increase the index by one and divide by the new index. For example, becomes .
The appears because the derivative of any constant term is zero.
C is called the (arbitrary) constant of integration. Its value can be found when appropriate additional information is provided, and this gives a particular integral.
The rule for integration is
This can also be written in the form provided .
In general or
Here's an example.
Find the equation of the curve for which and which passes through the point (1, 3).
substituting x = 1 and y = 3 gives 3 = 1 + 2 + C therefore C = 0
Find the equation of the curve for which and which passes through the point (2, 9).
substituting x = 2 and y = 9