What is integration?
- Integration is the opposite to differentiation
- Integration is referred to as antidifferentiation
- The result of integration is referred to as the antiderivative
- Integration is the process of finding the expression of a function (antiderivative) from an expression of the derivative (gradient function)
What is the notation for integration?
- An integral is normally written in the form
- the large operator means “integrate”
- “” indicates which variable to integrate with respect to
- is the function to be integrated (sometimes called the integrand)
- The antiderivative is sometimes denoted by
- there’s then no need to keep writing the whole integral; refer to it as
- may also be called the indefinite integral of
What is the constant of integration?
- Recall one of the special cases from Differentiating Powers of x
- If then
- This means that integrating 0 will produce a constant term in the antiderivative
- a zero term wouldn’t be written as part of a function
- every function, when integrated, potentially has a constant term
- This is called the constant of integration and is usually denoted by the letter
- it is often referred to as “plus ”
- Without more information it is impossible to deduce the value of this constant
- there are endless antiderivatives, , for a function
How do I integrate powers of x?
- Powers of are integrated according to the following formulae:
- If then where and is the constant of integration
- This is given in the formula booklet
- If the power of is multiplied by a constant then the integral is also multiplied by that constant
- If then where and is a constant and is the constant of integration
- notation can still be used with integration
- Note that the formulae above do not apply when as this would lead to division by zero
- Remember the special case:
- This allows constant terms to be integrated
- Functions involving roots will need to be rewritten as fractional powers of first
- eg. If then rewrite as and integrate
- Functions involving fractions with denominators in terms of will need to be rewritten as negative powers of first
- e.g. If then rewrite as and integrate
- The formulae for integrating powers of apply to all rational numbers so it is possible to integrate any expression that is a sum or difference of powers of
- e.g. If then
- e.g. If then
- Products and quotients cannot be integrated this way so would need expanding/simplifying first
- e.g. If then
What might I be asked to do once I’ve found the anti-derivative (integrated)?
- With more information the constant of integration,, can be found
- The area under a curve can be found using integration
- You can speed up the process of integration in the exam by committing the pattern of basic integration to memory
- In general you can think of it as 'raising the power by one and dividing by the new power'
- Practice this lots before your exam so that it comes quickly and naturally when doing more complicated integration questions
find an expression for in terms of.