Constants of Integration (Edexcel IGCSE Further Pure Maths): Revision Note

Finding the Constant of Integration

What is the constant of integration? 

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

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

  • $c$ 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

    • $\frac{\mathrm{d}}{\mathrm{d}x}\left(x^3-5x\right)=3x^2-5$

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

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

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

• 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 $\frac{\mathrm{d}y}{\mathrm{d}x}$ or ${\mathrm{f}}^{\text{'}}\left(x\right)$

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

    • $\int \frac{\mathrm{d}y}{\mathrm{d}x}\mathrm{d}x=y+c$

    • $\int {\mathrm{f}}^{\text{'}}\left(x\right)\mathrm{d}x=\mathrm{f}\left(x\right)+c$

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

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

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

    • 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 $\mathrm{f}\left(x\right)$ for one value of $x$

    • you can substitute and solve to find $c$