Initial condition and connection to the integral

Finding THE correct antiderivative

A function f has an infinite number of antiderivatives of the form F(x) + C. To determine ONE specific antiderivative, a condition of the form F(x₀) = y₀ (a specified value at a point) is generally given.

Method

1. Calculate the general antiderivative F(x) + C.
2. Replace x with x₀ and set up the equation F(x₀) + C = y₀.
3. Solve to find C.
4. Write the complete antiderivative.

Example

Let f(x) = 2x. The general antiderivatives are F(x) = x² + C. We are looking for the one that satisfies F(1) = 4. 1² + C = 4, so C = 3. The antiderivative we are looking for is F(x) = x² + 3.

Link to integration

Integral calculus relies directly on antiderivatives: if F is an antiderivative of f on [a; b], then the integral from a to b of f(x) dx = F(b) – F(a). This is the fundamental theorem of calculus, which links the area under the curve to antiderivatives.

Example

The integral from 0 to 2 of x² dx = [x³/3] between 0 and 2 = 8/3 – 0 = 8/3.

Common pitfall

Do not forget that the result F(b) – F(a) does NOT depend on the choice of the constant C: whichever antiderivative is chosen, the difference remains the same (the Cs cancel each other out). Also, be careful to get the order right: it is F(b) – F(a) and not the other way round.