Integral Formula

The Formula

\int f(x) \, dx = F(x) + C where F'(x) = f(x)

When to use: If derivative gives rate, integral gives total. Derivative of position = velocity; integral of velocity = position.

Quick Example

\int 2x \, dx = x^2 + C The area under f(x) = 2x from 0 to 3 is 9.

Notation

\int f(x)\,dx denotes the indefinite integral (antiderivative). F(x) is any antiderivative; C is the constant of integration.

What This Formula Means

The reverse operation of differentiation; it also computes the exact area under a curve between two points.

If derivative gives rate, integral gives total. Derivative of position = velocity; integral of velocity = position.

Formal View

F is an antiderivative of f on (a, b) if F'(x) = f(x) for all x \in (a, b). The indefinite integral: \int f(x)\,dx = \{F(x) + C : C \in \mathbb{R}\} where F' = f.

Worked Examples

Example 1

easy
Find \int (4x^3 + 6x) \, dx

Solution

  1. 1
    Apply the power rule for integration: \int x^n \, dx = \frac{x^{n+1}}{n+1} + C.
  2. 2
    For 4x^3: \frac{4x^4}{4} = x^4.
  3. 3
    For 6x: \frac{6x^2}{2} = 3x^2.
  4. 4
    Combine with the constant of integration: x^4 + 3x^2 + C.

Answer

x^4 + 3x^2 + C
Integration reverses differentiation. The power rule for integration adds 1 to the exponent and divides by the new exponent. Always include the constant C for indefinite integrals.

Example 2

medium
Evaluate \int_0^2 (3x^2 + 1) \, dx

Common Mistakes

  • Forgetting the constant of integration +C on indefinite integrals — without it, you have only one specific antiderivative, not the general solution.
  • Reversing the power rule incorrectly: \int x^n \, dx = \frac{x^{n+1}}{n+1} + C, not \frac{x^{n+1}}{n} or \frac{x^n}{n+1}.
  • Thinking the integral of \frac{1}{x} is \frac{x^0}{0} — the power rule doesn't apply when n = -1; the answer is \ln|x| + C.

Why This Formula Matters

Integration computes exact areas, volumes, and totals accumulated from known rates of change.

Frequently Asked Questions

What is the Integral formula?

The reverse operation of differentiation; it also computes the exact area under a curve between two points.

How do you use the Integral formula?

If derivative gives rate, integral gives total. Derivative of position = velocity; integral of velocity = position.

What do the symbols mean in the Integral formula?

\int f(x)\,dx denotes the indefinite integral (antiderivative). F(x) is any antiderivative; C is the constant of integration.

Why is the Integral formula important in Math?

Integration computes exact areas, volumes, and totals accumulated from known rates of change.

What do students get wrong about Integral?

Always write +C for indefinite integrals—omitting it loses the entire family of antiderivatives.

What should I learn before the Integral formula?

Before studying the Integral formula, you should understand: derivative.

Want the Full Guide?

This formula is covered in depth in our complete guide:

How to Integrate Rational Functions: Long Division and Partial Fractions →
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