Integral Formula

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

The Formula

โˆซf(x)โ€‰dx=F(x)+C\int f(x) \, dx = F(x) + C where Fโ€ฒ(x)=f(x)F'(x) = f(x)

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

Quick Example

โˆซ2xโ€‰dx=x2+C\int 2x \, dx = x^2 + C The area under f(x)=2xf(x) = 2x from 0 to 3 is 9.

Notation

โˆซf(x)โ€‰dx\int f(x)\,dx denotes the indefinite integral (antiderivative). F(x)F(x) is any antiderivative; CC 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

FF is an antiderivative of ff on (a,b)(a, b) if Fโ€ฒ(x)=f(x)F'(x) = f(x) for all xโˆˆ(a,b)x \in (a, b). The indefinite integral: โˆซf(x)โ€‰dx={F(x)+C:CโˆˆR}\int f(x)\,dx = \{F(x) + C : C \in \mathbb{R}\} where Fโ€ฒ=fF' = f.

Worked Examples

Example 1

easy
Find โˆซ(4x3+6x)โ€‰dx\int (4x^3 + 6x) \, dx

Answer

x4+3x2+Cx^4 + 3x^2 + C

First step

1
Apply the power rule for integration: โˆซxnโ€‰dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C.

Full solution

  1. 2
    For 4x34x^3: 4x44=x4\frac{4x^4}{4} = x^4.
  2. 3
    For 6x6x: 6x22=3x2\frac{6x^2}{2} = 3x^2.
  3. 4
    Combine with the constant of integration: x4+3x2+Cx^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 CC for indefinite integrals.

Example 2

medium
Evaluate โˆซ02(3x2+1)โ€‰dx\int_0^2 (3x^2 + 1) \, dx

Example 3

easy
Evaluate โˆซ032xโ€‰dx\int_0^3 2x \, dx.

Common Mistakes

  • Forgetting the +C+C on an indefinite integral โ€” many functions have the same derivative, so the constant must be carried.
  • Reversing the power rule wrong - โˆซxnโ€‰dx=xn+1n+1+C\int x^n\,dx=\frac{x^{n+1}}{n+1}+C (for nโ‰ โˆ’1n\ne -1) raises the power and divides, the opposite of differentiation; the case n=โˆ’1n=-1 gives lnโกโˆฃxโˆฃ+C\ln|x|+C instead.
  • Treating โˆซxโ€‰dx\int x\,dx as if it had a numeric answer โ€” 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+C is the classic error, and it reflects a deeper truth โ€” 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+C), rather than a numeric area?" โ€” rather than by familiar numbers โ€” 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?

โˆซf(x)โ€‰dx\int f(x)\,dx denotes the indefinite integral (antiderivative). F(x)F(x) is any antiderivative; CC 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+C is the classic error, and it reflects a deeper truth โ€” 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+C), rather than a numeric area?" โ€” rather than by familiar numbers โ€” 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+C on an indefinite integral. Asking "Am I looking for a function whose derivative is the given one (with a +C+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 โ†’
โ† Back to Integral See All Examples Practice Problems