Integration resources

A Calculus Refresher. This booklet revises techniques in calculus (differentiation and integration). This is a welsh language version

The second major component of the Calculus is called integration. This may be introduced as a means of finding areas using summation and limits. We shall adopt this approach in the present Unit. In later units, we shall also see how integration may be related to differentiation.

A special rule, integration by parts, is available for integrating products of two functions. This unit derives and illustrates this rule with a number of examples.

The derivative of ln 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.

Sometimes the integral of an algebraic fraction can be found by first expressing the algebraic fraction as the sum of its partial fractions. In this unit we will illustrate this idea. We will see that it is also necessary to draw upon a wide variety of other techniques such as completing the square, integration by substitution, using standard forms, and so on.

This unit explains how trig identities and trig substitutions can help when finding integrals.

Integration can be used to calculate areas. In simple cases, the area is given by a single definite integral. But sometimes the integral gives a negative answer which is minus the area, and in more complicated cases the correct answer can be obtained only by splitting the area into several parts and adding or subtracting the appropriate integrals. (Mathtutor Video Tutorial) This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Sometimes the integral of an algebraic fraction can be found by first expressing the algebraic fraction as the sum of its partial fractions. In this unit we look at the case where the denominator of the fraction involves an irreducible quadratic expression. (Mathtutor Video Tutorial) This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

This unit explain integration as the reverse of differentiation. (Mathtutor Video Tutorial) This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

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. When dealing with definite integrals, the limits of integration can also change. In this unit we will meet several examples of integrals where it is appropriate to make a substitution. (Mathtutor Video Tutorial) This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

We may regard integration as the reverse of differentiation. So if we have a table of derivatives, we can read it backwards as a table of anti-derivatives. When we do this, we often need to deal with constants which arise in the process of Differentiation. (Mathtutor Video Tutorial) This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

We sometimes need to calculate the volume of a solid which can be obtained by rotating a curve about the x-axis. There is a straightforward technique which enables this to be done, using integration. (Mathtutor Video Tutorial) This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Sometimes the integral of an algebraic fraction can be found by first expressing the algebraic fraction as the sum of its partial fractions. In this unit we will illustrate this idea. We will see that it is also necessary to draw upon a wide variety of other techniques such as completing the square, integration by substitution, using standard forms, and so on. (Mathtutor Video Tutorial) The video is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

The second major component of the Calculus is called integration. This may be introduced as a means of finding areas using summation and limits. We shall adopt this approach in the present Unit. In later units, we shall also see how integration may be related to differentiation. (Mathtutor Video Tutorial) The video is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

A special rule, integration by parts, is available for integrating products of two functions. This unit derives and illustrates this rule with a number of examples. (Mathtutor Video Tutorial) The video is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

This unit is concerned with integrals which lead to logarithms. Whenever the integrand is fraction with denominator f(x) and numerator f'(x) the result of integrating is the natural logarithm of f(x). This unit illustrates this behaviour with several examples. (Mathtutor Video Tutorial) The video is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

This unit explains how trig identities and trig substitutions can help when finding integrals. (Mathtutor Video Tutorial) The video is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.