Constants of Integration (Edexcel IGCSE Further Maths)

Finding the Constant of Integration

Spec point — 9A

What is the constant of integration?

• The constant of integration is the $c$ used when finding an indefinite integral

  • e.g. $\int ( 3 x^{2} - 5 ) \mathrm{d} x = x^{3} - 5 x + c$
  • $c$ 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

    • $\frac{ \mathrm{d} }{ \mathrm{d} x} ( x^{3} - 5 x ) = 3 x^{2} - 5$
    • $\frac{ \mathrm{d} }{ \mathrm{d} x} ( x^{3} - 5 x + 7 ) = 3 x^{2} - 5$
    • $\frac{ \mathrm{d} }{ \mathrm{d} x} ( x^{3} - 5 x - 498 ) = 3 x^{2} - 5$

  • So $x^{3}-5x$ plus or minus any constant is a valid solution to $\int ( 3 x^{2} - 5 ) \mathrm{d} x$

• But consider the graph of $y=x^{3}-5x+c$

  • Different values of $c$ represent different vertical translations of the graph
  • If we know one point on that graph we can work out the value of $c$

How do I find the constant of integration?

• On an exam you may be given a derivative $\frac{dy}{dx}$ or $\mathrm{f}^{'} ( x )$

  • You can integrate that to find $y$ or $\mathrm{f} ( x )$ in '$+c$' form

    • $\int \frac{ \mathrm{d} y}{ \mathrm{d} x} \mathrm{d} x = y + c$
    • $\int \mathrm{f}^{'} ( x ) \mathrm{d} x = \mathrm{f} ( x ) + c$

• If you are also given a point on the graph of $y$ or of $y = \mathrm{f} ( x )$

  • you can use this to find the value of $c$

  • Substitute the values you know into $y+c$ or $\mathrm{f} ( x ) + c$

    • Then solve for $c$

• The extra information doesn't have to be a point on a graph

  • As long as you know the value of $\mathrm{f} ( x )$ for one value of $x$

    • you can substitute and solve to find $c$

Exam Hint

• 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!

Worked Example The graph of $y=f(x)$ passes through the point $(3,-4)$. The derivative of $f(x)$ is given by $f^{'}(x)=3x^{2}-4x-4$.

Find $f(x)$.

Integrate $\mathrm{f}^{'} ( x )$ to find $\mathrm{f} ( x )$ in '$+c$' form

$\begin{gathered} \mathrm{f} ( x ) & = \int ( 3 x^{2} - 4 x - 4 ) \mathrm{d} x \\ = 3 ( \frac{x^{2 + 1} }{2 + 1} ) - 4 ( \frac{x^{1 + 1} }{1 + 1} ) - 4 x + c \\ = x^{3} - 2 x^{2} - 4 x + c \end{gathered}$

We also know the curve of $y = \mathrm{f} ( x )$ goes through $( 3 , - 4 )$

That means the function is equal to $-4$ when $x=3$

$( 3 )^{3} - 2 ( 3 )^{2} - 4 ( 3 ) + c = - 4$

Solve for $c$

$\begin{gathered} 27 - 18 - 12 + c & = - 4 \\ c - 3 & = - 4 \\ c = - 1 \end{gathered}$

$f(x)=x^{3}-2x^{2}-4x-1$