Constant of Integration (College Board AP® Calculus BC): Revision Note

Finding the constant of integration

How can I find the value of a constant of integration?

• When finding an indefinite integral, a constant of integration is needed

  • $\int f\left(x\right)dx=F\left(x\right)+C$

    • where $F^{\text{'}}\left(x\right)=f\left(x\right)$

    • and $C$ is any constant

• If you know more information about $F\left(x\right)$ you can work out the value of the constant $C$

  • You may be given the value of $F\left(x\right)$ for some particular value of $x$

  • Or you may be told that the graph of $F\left(x\right)$ goes through a particular point $\left(x_0,y_0\right)$

    • Remember in this case that $y_0=F\left(x_0\right)$

  • This lets you set up and solve an equation to find the value of $C$

• For example, if the function $F$ satisfies $F\text{'}\left(x\right)=2x+3$, and you also know that the graph of $F$ goes through the point $\left(1,2\right)$

  • First integrate

    • $F\left(x\right)=\int \left(2x+3\right)dx=x^2+3x+C$

  • The graph goes through $\left(1,2\right)$, so $F\left(1\right)=2$

    • $F\left(1\right)={\left(1\right)}^2+3\left(1\right)+C=2$

  • Solve the equation for $C$

    • $\begin{array}{rcl}1+3+C& =& 2\Rightarrow C+4=2\Rightarrow C=-2\end{array}$

  • Therefore

    • $F\left(x\right)=x^2+3x-2$

• Finding the constant of integration in this way is equivalent to finding the particular solution of a first-order differential equation in the form $\frac{dy}{dx}=f\left(x\right)$

  • See the 'Particular Solutions' study guide for a more formal treatment of this