Introduction to Integration (Edexcel IGCSE Further Maths)

Spec point — 9A

Introduction to Integration

What is 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 $\int \mathrm{f}^{'} ( x ) \mathrm{d} x = \mathrm{f} ( x ) + c$

  • $\int ... \mathrm{d} x$ is the integral with respect to$x$ of "..."

  • $c$ 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

Exam Hint

• 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 ($+c$) when finding an indefinite integral

  • Leaving it out can lose marks

Worked Example (a) Show that the derivative of $x^{3}+\frac{1}{x}$ is $3x^{2}-\frac{1}{x^{2}}$.

This is a standard 'powers of $x$'derivative Just remember to rewrite $\frac{1}{x}$ as a power using laws of indices

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

$\begin{gathered} \mathrm{f}^{'} ( x ) & = 3 x^{3 - 1} + ( ( - 1 ) x^{- 1 - 1} ) \\ = 3 x^{2} - x^{- 2} \end{gathered}$

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

$\begin{array}{ccc} f^{ } \text{'} \text{stretchy} ( x \text{stretchy} ) & = & 3 x^{ } 2 - \frac{1}{ } x^{ } 2\end{array}$

(b) Hence write down the answer to the indefinite integral $\int ( 3 x^{2} - \frac{1}{x^{2}} ) \mathrm{d} x$.

We can use $\int \mathrm{f}^{'} ( x ) d x = \mathrm{f} ( x ) + c$ to write down the answer

This is because integration and differentiation are inverse operations

Just don't forget the constant of integration $+c$

$\text{integral} \text{stretchy} ( 3 x ^{2} - \frac{1}{x} ^{2} \text{stretchy} ) d x = x^{ } 3 + \frac{1}{ } x + c$