Introduction to Integration

Integration is the inverse operation to differentiation
- So if you differentiate a function to find its derivative
- and then integrate that derivative
- you should end up back at the original function
This can be written as
- $\int . . . d x$ is the integral with respect to $x$ of "..."
- is the constant of integration
  - A derivative gives the rate of change of a function
    - but it doesn't give the starting point for any change
  - The constant of integration represents this 'starting point'
  - See the 'Constants of Integration' revision note for more info
This type of integral is known as an indefinite integral
- The answer to an indefinite integral is another function
  - There are also definite integrals
    - The answer to a definite integral is a number
  - See the 'Calculating Areas' revision note for more info on definite integrals
Usually you will integrate using standard formulae
- See the 'Integrating Basic Functions' revision note for these formulae

Remember that integration and differentiation are inverse operations
- So if you differentiate your answer to an indefinite integral
  - you should end up back at the function you were integrating
- Use this to check your answers on the exam!
Don't forget the constant of integration () when finding an indefinite integral
- Leaving it out can lose marks

(a)

Show that the derivative of

x^{3} + \frac{1}{x}

3 x^{2} - \frac{1}{x^{2}}

This is a standard 'powers of

x

'derivative
Just remember to rewrite

\frac{1}{x}

as a power using laws of indices

f (x) = x^{3} + x^{- 1}

\begin{array}{rcl} f^{'} (x) & = & 3 x^{3 - 1} + ((- 1) x^{- 1 - 1}) \\ = & 3 x^{2} - x^{- 2} \end{array}

Now just use laws of indices again to write the answer in the requested form

\begin{array}{rcl} f^{'} (x) & = & 3 x^{2} - \frac{1}{x^{2}} \end{array}

(b)

Hence write down the answer to the indefinite integral

\int (3 x^{2} - \frac{1}{x^{2}}) d x

We can use

\int f^{'} (x) d x = f (x) + c

to write down the answer

This is because integration and differentiation are inverse operations

Just don't forget the constant of integration

+ c

\int (3 x^{2} - \frac{1}{x^{2}}) d x = x^{3} + \frac{1}{x} + c

Introduction to Integration (Edexcel IGCSE Further Maths)