Integration

Integration as a summation

Integration as summation . Integration may be introduced as a means of finding areas using summation and limits. This process gives rise to the definite integral of a function.

Integration as the reverse of differentiation

Integration as the reverse of differentiation. Integration can be seen as differentiation in reverse; that is we start with a given function f(x), and ask which functions, F(x), would have f(x) as their derivative. The result is called an indefinite integral. A definite integral can be obtained by substituting values into the indefinite integral.

Integration using a table of anti-derivatives

Integration using a table of anti-derivatives. Integration may be regarded as the reverse of differentiation, so a table of derivatives can be read backwards as a table of anti-derivatives. The final result for an indefinite integral must, however, include an arbitrary constant.

Integration by Parts

Integration by parts. A special rule, integration by parts, can often be used to integrate the product of two functions. It is appropriate when one of the functions forming the product is recognised as the derivative of another function. The result still involves an integral, but in many cases the new integral will be simpler than the original one.

Integration by substitution

Integration by substitution . There are occasions when it is possible to perform an apparently difficult piece of integration by first making a substitution. This has the effect of changing the variable and the integrand. With definite integrals the limits of integration can also change.

Integration using Algebraic Fractions Part 1

Integrating algebraic fractions (1). The integral of an algebraic fraction can often be found by first expressing the fraction as the sum of its partial fractions. To do this it is necessary to draw on a wide variety of other techniques. This unit considers the case where the denominator may be written as a product of linear factors.

Integration using Algebraic Fractions Part 2

Integrating algebraic fractions (2). The integral of an algebraic fraction can often be found by first expressing the fraction as the sum of its partial fractions. Further techniques are available when the denominator involves an irreducible quadratic expression.

Integration using Trigonometric Formulae

Integration using trigonometric formulae. Sometimes integrals involving trigonometric functions can be evaluated by using trigonometric identities. These allow the integrand to be written in an alternative form which may be more amenable to integration. Trigonometric formulae can also be used in substitutions to simplify complicated integrals.

Finding areas by integration

Finding areas by integration. In simple cases, areas can be found by evaluating a single definite integral. Sometimes the integral gives a negative answer, and sometimes the correct answer can be obtained only by splitting the area into several parts and adding and subtracting appropriately.

Volumes of solids of revolution

Volumes of solids of revolution. A solid of revolution is obtained by rotating a curve about the x-axis. There is a straightforward technique, using integration, which enables us to calculate the volume of such a solid.

Integration leading to log functions

Integration leading to log functions . The derivative of x is 1/x. As a consequence, if we reverse the process, the integral of 1/x is ln x + c. In this unit we generalise this result and see how a wide variety of integrals result in logarithm functions.