Finding the Constant of Integration
What is the constant of integration?
- The constant of integration is the used when finding an indefinite integral
- e.g.
- can be any constant
- Integration and differentiation are inverse operations
- That means if you differentiate your answer to an integral...
- ...it should turn back into the original function you integrated
- The 'problem' is that the derivative of a constant is zero
- So plus or minus any constant is a valid solution to
- That means if you differentiate your answer to an integral...
- But consider the graph of
- Different values of represent different vertical translations of the graph
- If we know one point on that graph we can work out the value of
How do I find the constant of integration?
- On an exam you may be given a derivative or
- You can integrate that to find or in '' form
- You can integrate that to find or in '' form
- If you are also given a point on the graph of or of
- you can use this to find the value of
- Substitute the values you know into or
- Then solve for
- The extra information doesn't have to be a point on a graph
- As long as you know the value of for one value of
- you can substitute and solve to find
- As long as you know the value of for one value of
Exam Tip
- An exam question probably won't tell you to 'find the constant of integration'
- Instead you'll be given the derivative of a function
- and one value of the function or a point on its graph
- Be sure to recognise this as a 'constant of integration' question!
- Instead you'll be given the derivative of a function
Worked example
The graph of passes through the point . The derivative of is given by .
Find .
Integrate to find in '' form
We also know the curve of goes through
That means the function is equal to when
Solve for