Indefinite Integrals

If F(x) is a function whose derivative F'(x) = f(x) on certain interval of the x-axis, then F(x) is called the anti-derivative of indefinite integral f(x). When we integrate the differential of a function we get that function plus an arbitrary constant. In symbols we write

$\displaystyle \int f (x)\,dx = F(x) + C$


where the symbol $\displaystyle \int$, called the integral sign, specifies the operation of integration upon f(x) dx; that is, we are to find a function whose derivative is f(x) or whose differential is f(x) dx. The dx tells us that the variable of integration is x.

Properties of Integrals

Integration Formulas
In these formulas, u and v denote differentiable functions of some independent variable (say x) and a, n, and C are constants.

  1. The integral of the differential of a function u is u plus an arbitrary constant C (the definition of an integral).

    $$\displaystyle \int du = u + C$$

  2. The integral of a constant times the differential of the function. (A constant may be written before the integral sign but not a variable factor).

    $$\displaystyle \int a \, du = a\int du$$

  3. The integral of the sum of a finite number of differentials is the sum of their integrals.

    $$\displaystyle \int (du + dv + ... + dz) = \int du + \int dv + ... + \int dz$$

  4. If n is not equal to minus one, the integral of un du is obtained by adding one to the exponent and divided by the new exponent. This is called the General Power Formula.

    $$\displaystyle \int u^n \, du = \dfrac{u^{n + 1}}{n + 1} + C; \, n \neq -1$$


Definite Integral

The definite integral of f(x) is the difference between two values of the integral of f(x) for two distinct values of the variable x. If the integral of f(x) dx = F(x) + C, the definite integral is denoted by the symbol

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


The quantity F(b) - F(a) is called the definite integral of f(x) between the limits a and b or simply the definite integral from a to b. It is called the definite integral because the result involves neither x nor the constant C and therefore has a definite value. The numbers a and b are called the limits of integration, a being the lower limit and b the upper limit.

General Properties of Definite Integral

  1. The sign of the integral changes if the limits are interchanged.

    $$\displaystyle \int_a^b f(x) \, dx = -\int_b^a f(x) \, dx$$

  2. The interval of integration may be broken up into any number of sub-intervals, and integrate over each interval separately.

    $$\displaystyle \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx$$

  3. The definite integral of a given integrand is independent of the variable of integration. Hence, it makes no difference what letter is used for the variable of integration.

    $$\displaystyle \int_a^b f(x) \, dx = \int_a^b f(z) \, dz$$