How Do You Find the Constant of Integration Using a Boundary Condition?

You find the constant of integration by first integrating, then substituting the values supplied by the boundary condition into the resulting equation and solving for $C$.

This is part of IB Mathematics: Applications and Interpretation SL 5.5: Introduction to integration. If

$$\frac{dy}{dx}=f(x),$$

then integrating gives

$$y=\int f(x)\,dx=F(x)+C.$$

The derivative does not reveal the original vertical position of the function because every function $F(x)+C$ has the same derivative. A boundary condition, such as $y=7$ when $x=2$, identifies one particular function from this family.

The boundary condition is a known point on the curve, not another differentiation rule. Substitution converts the unknown vertical shift into a simple equation. If the condition is written as $y(a)=b$, substitute $x=a$ and $y=b$ into $y=F(x$, giving .

Use this reliable sequence: (1) integrate every term, (2) include $+C$, (3) translate the condition into coordinates, (4) substitute both coordinates, and (5) solve the resulting linear equation for $C$. Finally, rewrite the complete function. In AI, a GDC may check arithmetic, but it should not replace the mathematical setup that demonstrates how the boundary condition determines the constant.

This distinction matters because the boundary condition selects one antiderivative, whereas limits calculate a net change or signed area.

Worked Example

Suppose

$$\frac{dy}{dx}=6x-4$$

and the curve passes through $(2,7)$.

First, integrate:

$$y=3x^2-4x+C.$$

Next, use $x=2$ and $y=7$:

$$7=3(2)^2-4(2)+C=4+C.$$

Therefore, $C=3$, so

$$y=3x^2-4x+3.$$

Verify by differentiating and by substituting $x=2$.

A common misconception is to omit $+C$ before applying the condition. This is incorrect because an indefinite integral represents a family of functions.

IB Exam Technique

Show three stages: integrate with $+C$, substitute the boundary condition, and solve for $C$. This working provides essential method evidence on either AI SL paper. Keep exact values unless rounding is requested.