**Indefinite integration** means *antidifferentiation*; that is, given a function ƒ( *x*), determine the most general function *F*( *x*) whose derivative is ƒ ( *x*). The symbol for this operation is the integral sign, ∫, followed by the **integrand** (the function to be integrated) and differential, such as *dx*, which specifies the variable of integration.

On the other hand, the fundamental geometric interpretation of **Definite integration** is to *compute an area*. That is, given a function ƒ( *x*) and an interval *a* ≤ *x* ≤ *b* in its domain, the definite integral of ƒ from *a* to *b* gives the area bounded by the curvey *y* = ƒ ( *x*), the *x* axis, and the vertical lines *x* = *a* and *x* = *b*. The symbol for this operation is the integral sign with **limits of integration** ( *a* and *b*), ƒ _{a }^{b }, followed by the function and the differential which specifies the variable of integration.

**Figure 1**

From their definitions, you can see that the processes of indefinite integration and definite integration are really very different. The *indefinite* integral of a function is the collection of functions which are its antiderivatives, whereas the *definite* integral of a function requires two limits of integration and gives a numerical result equal to an area in the *xy* plane. However, the fact that both operations are called “integration” and are denoted by such similar symbols suggests that there is a link between them.

The **Fundamental Theorem of Calculus** says that differentiation (finding the slope of a curve) is the inverse operation of definite integration (finding the area under a curve). More explicitly, Part **I** of the Fundamental Theorem says that if a function is integrated (to form a definite integral with a variable upper limit of integration), and the result is then differentiated, the original function is recovered; that is, differentiation “undoes” integration. Part **II** gives the connection between definite and indefinite integrals. It says that a definite integral can be computed by first determining an indefinite integral (so computing the area under a curve is done by antidifferentiating).

*The Fundamental Theorem of Calculus (Part I*):

*If*i> ƒ is continuous, then *d* *dx* ∫ ^{x }_{a }ƒ( *t*) *dt* = ƒ( *x*).

*The Fundamental Theorem of Calculus (Part II)*:

*If* ƒ is continuous with antiderivative *F*, then ∫ ^{b }_{a }ƒ( *x*) *dx* = *F*( *b*) − *F*( *a*).

**Example 1**: Evaluate the integral

Using the first integration formula in Table 1, every function whose derivative equals ƒ( *x*) = *x* ^{4} − 3 *x* ^{2} + *x* − 1 is given by

where *c* is an arbitrary constant.

**TABLE 1 ****Differentiation and Integration Formulas**

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