Finding the Constant of Integration (DP IB Analysis & Approaches (AA): HL): Revision Note

Finding the constant of integration

What is the constant of integration? 

• When finding an anti-derivative there is a constant term to consider

  • This constant term, usually called $c$, is the constant of integration

• In terms of graphing an anti-derivative, there are endless possibilities

  • Collectively these may be referred to as the family of antiderivatives or family of curves

    • They all have the same shape, but are just shifted up or down vertically compared to each other

    • Each one corresponds to a different possible value of $c$

• The value of the constant of integration is determined by the exact location of the curve

  • if a point on the curve is known, the constant of integration can be found

How do I find the constant of integration?

• For $\int f\left(x\right)\mathrm{d}x=\mathrm{F}\left(x\right)+c$, the constant of integration, $c$, can be found if a point on the graph of $y=\mathrm{F}\left(x\right)+c$ is known 
   

• STEP 1

   If need be, rewrite$f\left(x\right)$ into an integrable form

  Each term needs to be a power of $x$ (or a constant)

   

• STEP 2

   Integrate each term of$f\left(x\right)$, remembering the constant of integration “$+c$”

    

• STEP 3

  Substitute the $x$ and $y$ coordinates of a given point in to $\mathrm{F}\left(x\right)+c$ to form an equation in $c$

  Solve the equation to find $c$