## 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

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.

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

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

- 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).

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

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

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

- If
*n*is not equal to minus one, the integral of*u*is obtained by adding one to the exponent and divided by the new exponent. This is called the^{n}du**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

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

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

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

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

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

- 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.

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