# 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

### 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 (21)

Diagnostic Test - Integration - Finding volumes

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Diagnostic Test - Integrating Algebraic Fractions 1

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Diagnostic Test - Integrating Algebraic Fractions 2

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Diagnostic Test - Integration - Finding areas

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Diagnostic Test - Integration as a summation

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Diagnostic Test - Integration as the reverse of Differentiation

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Diagnostic Test - Integration by parts

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Diagnostic Test - Integration requiring use of trigonometric identities

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Diagnostic Test - Integration that leads to log functions

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Diagnostic Test - Integration using a table and standard rules

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Diagnostic Text - Integration by substitution

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Exercise - Integrating algebraic fractions 1

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Exercise - Integrating algebraic fractions 2

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Exercise - Integration - Finding areas

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

Exercise - Integration as the reverse of Differentiation

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

Exercise - Integration by parts

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

Exercise - Integration by substitution

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

Exercise - Integration requiring use of trigonometric identities

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

Exercise - Integration that leads to log functions

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

Exercise - Integration to find volumes

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

Exercise - Integration using a table and standard rules

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### 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.