Indefinite Integral

An indefinite integral represents the set of all antiderivatives F(x) + k of a continuous function f(x) over an interval [a, b]. It is denoted as $$ \int f(x) \: dx = F(x) + k $$ where F(x) is any antiderivative of f(x), and k is an arbitrary constant.

The indefinite integral is the reverse process of the derivative for continuous functions.

A practical example

Consider the function f(x) = 2x. Its indefinite integral is:

$$ \int 2x \:dx = x^2 + k $$

The expression x² + k represents the complete family of antiderivatives of f(x):

$$ x^2 + k = \begin{cases} x^2 - 3 \\ x^2 - 2 \\ x^2 - 1 \\ x^2 \\ x^2 + 1 \\ x^2 + 2 \\ \vdots \end{cases} $$

The derivative of any member of this family, F(x) + k, always returns the original function f(x):

$$ D[F(x) + k] = f(x) = 2x $$

This holds because the derivative of a constant is zero:

$$ D[F(x) + k] = D[f(x)] + D[k] = f(x) + 0 = 2x $$

Note: Here are a few sample antiderivatives of f(x): $$ F(x) = x^2 + 3 \\ F(x) = x^2 - 5 \\ F(x) = x^2 $$ In all cases, the derivative gives back f(x) = 2x, since the constant term disappears upon differentiation: $$ D[x^2 + 3] = 2x \\ D[x^2 - 3] = 2x \\ D[x^2] = 2x $$

More examples of basic integrals

$$ \int x^n \: dx = \frac{x^{n+1}}{n+1} + k \quad \text{(for } n \ne -1\text{)} $$

$$ \int \cos x \: dx = \sin x + k $$

$$ \int \sin x \: dx = - \cos x + k $$

$$ \int \frac{1}{x} \: dx = \log x + k $$

$$ \int e^x \: dx = e^x + k $$

Difference between definite and indefinite integrals

A definite integral yields a real number representing the area between the graph of a function and the x-axis over a given interval:

$$ \int_a^b f(x) \: dx $$

In contrast, an indefinite integral returns a family of antiderivatives:

$$ \int f(x) \: dx $$

Despite their differences, the two are connected by the Fundamental Theorem of Calculus:

$$ \int_a^b f(x) \: dx = F(b) - F(a) $$

Properties of Indefinite Integrals

Indefinite integrals follow the same algebraic rules as definite integrals:

• Sum Rule

  $$ \int [f(x) + g(x)] \: dx = \int f(x) \: dx + \int g(x) \: dx $$

  The derivative of a sum is the sum of the derivatives. Since integration reverses differentiation, the same property holds for integrals.
• Difference Rule

  $$ \int [f(x) - g(x)] \: dx = \int f(x) \: dx - \int g(x) \: dx $$

• Scalar Multiplication (Linearity)

  $$ \int k \cdot f(x) \: dx = k \cdot \int f(x) \: dx $$

And so on.