Integral Formula

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

The Formula

∫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

∫2x dx=x2+C The area under f(x)=2x from 0 to 3 is 9.

Notation

∫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∈(a,b). The indefinite integral: ∫f(x) dx={F(x)+C:C∈R} where F′=f.

Worked Examples

Example 1

easy
Find ∫(4x3+6x) dx

Answer

x4+3x2+C

First step

1
Apply the power rule for integration: ∫xn dx=xn+1n+1+C.

Full solution

  1. 2
    For 4x3: 4x44=x4.
  2. 3
    For 6x: 6x22=3x2.
  3. 4
    Combine with the constant of integration: x4+3x2+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 ∫02(3x2+1) dx

Example 3

easy
Evaluate ∫032x dx.

Common Mistakes

  • Forgetting the +C on an indefinite integral — many functions have the same derivative, so the constant must be carried.
  • Reversing the power rule wrong - ∫xn dx=xn+1n+1+C (for n≠−1) raises the power and divides, the opposite of differentiation; the case n=−1 gives ln⁡∣x∣+C instead.
  • Treating ∫x dx as if it had a numeric answer — without bounds an indefinite integral is a function, not a number.

Why This Formula Matters

Integration is how rates become totals: from velocity you recover position, from a marginal rate you recover the whole amount. Forgetting the +C is the classic error, and it reflects a deeper truth — infinitely many functions share the same derivative, so the antiderivative is a family, not a single function. Recognizing it by "Am I looking for a function whose derivative is the given one (with a +C), rather than a numeric area?" — rather than by familiar numbers — is what lets a student tell it apart from definite integral and derivative and riemann sum in a mixed problem set.

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?

∫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 is how rates become totals: from velocity you recover position, from a marginal rate you recover the whole amount. Forgetting the +C is the classic error, and it reflects a deeper truth — infinitely many functions share the same derivative, so the antiderivative is a family, not a single function. Recognizing it by "Am I looking for a function whose derivative is the given one (with a +C), rather than a numeric area?" — rather than by familiar numbers — is what lets a student tell it apart from definite integral and derivative and riemann sum in a mixed problem set.

What do students get wrong about Integral?

The procedure for integral is the easy part; the trap is forgetting the +C on an indefinite integral. Asking "Am I looking for a function whose derivative is the given one (with a +C), rather than a numeric area?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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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