Integrating Basic Functions (Edexcel IGCSE Further Pure Maths)
Revision Note
Written by: Roger B
Reviewed by: Dan Finlay
Integrating Powers of x
How do I integrate powers of x?
Powers of are integrated according to the following formula:
(where is the constant of integration)
This is valid for any value of except
So you cannot integrate this way
If is multiplied by a constant then
This also is not valid for
These formulae are not on the exam formula sheet, so you need to remember them
Remember the special case:
e.g.
This allows constant terms to be integrated
Functions involving roots will need to be rewritten as fractional powers
e.g.
rewrite as
then integrate
Fractions with in the denominator will need to be rewritten as negative powers
e.g.
rewrite as
then integrate
How do I integrate sums and differences of powers of x?
The formulae can be used to integrate sums or differences of powers of
Just integrate term by term
e.g.
Products and quotients cannot be integrated this way
You need to expand and/or simplify first
e.g.
expand as
then integrate term by term
you cannot just multiply the integrals of and together
What might I be asked to do once I’ve integrated?
You may be given the derivative of a function and asked to find the function
Integration and differentiation are inverse operations so
With more information the constant of integration,, can be found
The area under a curve can also be found using integration
Examiner Tips and Tricks
Remember the basic pattern of integrating powers of x
'Raise the power by one and divide by the new power'
Lots of practice will improve your speed and accuracy
It's easy to check your answer when integrating
Just differentiate your answer
It should turn back into the function you were integrating in the first place
Worked Example
Given that , find an expression for in terms of.
Start by rewriting entirely in powers of
By laws of indices
Remember
We can integrate term by term using
We can't find the value of without further info, so that's the answer to the question
It's 'nice' to turn back into for the final answer, but you would also get the marks without doing that
Integrating Trig Functions
How do I integrate sin and cos?
You can integrate and by using the formulae
is the constant of integration
If is multiplied by a constant then
None of these formulae are on the exam formula sheet, so you need to remember them
For calculus with trigonometric functions angles must be measured in radians
Make sure you know how to change the angle mode on your calculator
Examiner Tips and Tricks
Remember to include 'c', the constant of integration, for any indefinite integrals
As soon as you see a question involving integration and trigonometry
put your calculator into radians mode
Worked Example
Given that , find an expression for in terms of .
Remember that
We can integrate term by term using the integration formulae for and
We can't find the value of without further info, so that's the answer to the question
Integrating e^x
How do I integrate exponentials?
can be integrated using the formula
is the constant of integration
If is multiplied by a constant then
These formulae are not on the exam formula sheet, so you need to remember them
Examiner Tips and Tricks
Because ' is its own integral' it is quite easy to integrate exponentials
Just be careful dealing with the constant in
Worked Example
Given that , find an expression for in terms of .
Remember that
We can integrate term by term using the integration formula for
You might find it easier to split the fraction into two separate fractions first
We can't find the value of without further info, so that's the answer to the question
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