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Antiderivatives

The process of "Undoing" a Derivative.

In calculus, many operations come in pairs of opposites. Addition has subtraction, multiplication has division, and differentiation has antidifferentiation. An antiderivative is a function that reverses what the derivative did.

1. Definition and Notation

If we have a function f(x), its antiderivative is a function F(x) such that F'(x) = f(x).

We use the Indefinite Integral symbol (an elongated S) to represent this operation.

Notation:
∫ f(x) dx = F(x) + C

This is read as "The integral of f(x) with respect to x."

2. The Mystery of "+ C"

You will notice that every indefinite integral ends with "+ C". This stands for the Constant of Integration.

Why do we need it? Because the derivative of any constant number is zero.

  • Derivative of x² is 2x.
  • Derivative of x² + 5 is 2x.
  • Derivative of x² - 100 is 2x.

When we reverse the process (integrate 2x), we know the answer involves x², but we have lost the information about the constant. It could be any vertical shift of the graph.

[Image of family of curves for antiderivative x squared plus c]

Mathematically, we say that the antiderivative is not a single function, but a family of functions.

3. The Reverse Power Rule

Just as the Power Rule is the shortcut for derivatives, the Reverse Power Rule is the shortcut for integrals.

Instead of "Multiply then Subtract", we do the opposite: "Add then Divide".

[Image of power rule for integration formula]
Formula:
∫ xⁿ dx = (xⁿ⁺¹) / (n+1) + C
(Provided n ≠ -1)

Example:

Find the integral of x³.

  1. Add 1 to the exponent: 3 + 1 = 4.
  2. Divide by the new exponent: 4.
  3. Add C.
Answer: (x⁴ / 4) + C

4. Common Antiderivatives

You should memorize these standard reverse operations:

  • ∫ cos(x) dx = sin(x) + C (Because derivative of sin is cos)
  • ∫ sin(x) dx = -cos(x) + C (Because derivative of cos is -sin)
  • ∫ (1/x) dx = ln|x| + C (This is the special case where power rule fails)
  • ∫ eˣ dx = eˣ + C (The easiest one!)

5. Linearity

Like derivatives, integrals are linear. This means you can split them up by addition and pull out constant multipliers.

∫ (3x² + 4x) dx
= 3 ∫ x² dx + 4 ∫ x dx
= 3(x³/3) + 4(x²/2) + C
= x³ + 2x² + C