# Integration resources

### Practice & Revision (2)

Cwrs Gloywi Calcwlws

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

### Quick Reference (9)

Evaluating definite integrals

This leaflet explains what is meant by a definite integral and how it can be evaluated. (Engineering Maths First Aid Kit 8.9)

Integration as summation

An integral is defined as an infinite sum. This leaflet explains how this is done. This notion is important when we want to apply integration in many fields. (Engineering Maths First Aid Kit 8.12)

Integration as the reverse of differentiation

Integration is introduced here as the reverse of differentiation. (Engineering Maths First Aid Kit 8.6)

Integration by parts

This leaflet explains integration by parts. This is a technique for integrating a product of two functions (two functions multiplied together). (Engineering Maths First Aid Kit 8.10)

Integration by substitution

Some integrals can be evaluated by making an appropriate substitution to change the variable. This leaflet explains how this can be done. (Engineering Maths First Aid Kit 8.11)

Integration: Laplace Transforms

Reviews the techniques of integration needed to find and manipulate Laplace Transforms. This Quick Reference leaflet is contributed to the mathcentre Community Project by Leslie Fletcher and reviewed by Martin Randles, Liverpool John Moores University.

Linearity rules of integration

This leaflet explains how to integrate the sum of two functions, and constant multiples of functions, using 'linearity rules'. (Engineering Maths First Aid Kit 8.8)

Numerical Quadrature

This resource covering the Trapezium Rule, Simpsons Rule and an overview of error function has been contributed to the mathcentre Community Project by Josh Simpson and reviewed by Leslie Fletcher, Liverpool John Moores University.

Table of integrals

This leaflet provides a table of integrals of common functions. (Engineering Maths First Aid Kit 8.7)

### Teach Yourself (11)

Finding areas by integration

This unit looks at
how to calculate the area bounded by a curve using integration.

Integration as a summation

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.

Integration as the reverse of differentiation

This unit explain integration as the reverse of differentiation.

Integration by parts

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.

Integration that leads to logarithms

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.

Integration using a table of anti-derivatives

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.

Integration using partial fractions 1

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.

Integration using partial fractions 2

In this unit we are going to look at how we can integrate some more algebraic fractions. We shall concentrate on the case where the denominator of the fraction involves an irreducible quadratic factor. The case where all the factors of the denominator are linear has been covered in the first unit on integration using partial fractions.

Integration using trig identities and trig substitutions.

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

Volumes of solids of revolution

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. This unit will explain how.

### Test Yourself (4)

Integration using partial fractions - Numbas

4 questions on using partial fractions to solve indefinite integrals. Numbas resources have been made available under a Creative Commons licence by Bill Foster and Christian Perfect, School of Mathematics & Statistics at Newcastle University.

Diagnostic test in basic calculus - Numbas

12 calculus questions, differentiation and integration. Useful for self diagnosis. Numbas resources have been made available under a Creative Commons licence by Bill Foster and Christian Perfect, School of Mathematics & Statistics at Newcastle University.

Indefinite integration by substitution - Numbas

5 questions on using substitution to find indefinite integrals. Numbas resources have been made available under a Creative Commons licence by Bill Foster and Christian Perfect, School of Mathematics & Statistics at Newcastle University.

Integration by Parts - Numbas

4 questions on integrating by parts. Numbas resources have been made available under a Creative Commons licence by Bill Foster and Christian Perfect, School of Mathematics & Statistics at Newcastle University.

### Third Party Resources (1)

University of East Anglia (UEA) Interactive Mathematics and Statistics Resources

The Learning Enhancement Team at the University of East Anglia (UEA) has developed la series of interactive resources accessible via Prezi mind maps : Steps into Numeracy, Steps into Algebra, Steps into Trigonometry, Bridging between Algebra and Calculus, Steps into Calculus, Steps into Differential Equations, Steps into Statistics and Other Essential Skills.

### Video (11)

Finding areas by integration

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.

Integrating algebraic fractions (1)

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)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Integrating algebraic fractions (2)

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.

Integration as a summation

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)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Integration as the reverse of differentiation

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.

Integration by parts

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)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

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. 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.

Integration that leads to log functions

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)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Integration using a table of anti-derivatives

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.

Integration using trig identities or a trig substitution

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

Volumes of solids of revolution

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.