Integration Key Facts

Integration Key FactsFacts which are in the formula bookStandard Integrals you need to learn:

Further generalisations using the above integrals:

Area under a curve, where y = f(x) between x=a and x=b:

Volume of revolution formed by rotating y about the x axis, where y = f(x) between x=a and x=b:

General patterns you should remember:

Integrating trigonometric powers

(double angle formula used)

(double angle formula used)

Integrating using Addition Formula

To integrate sin(ax)sin(bx), cos(ax)cos(bx) cos (A (B) = cosAcosB ( sinAsinB (C3 section) are required.To integrate sin(ax)cos(bx) sin (A (B) = sinAcosB ( sinAcosB (C3 section) are required.Other trigonometric integrals:

Trapezium Rule (C2 section)

Separating variables:

Integrating parametric equations:When x = f(t), y = (t), the area under the curve:

Volume of revolution formed by rotating y about the x axis,

Integration By Parts

How do you know which is u and ?Integration by parts is used to integrate a product. One part of the product (u) should be easy to differentiate (and will usually simplify when differentiated). The other part should be easy to integrate and should not become too much harder when integrated.

Examples:Function to be integratedu needs differentiating to find

needs integrating to find v

x sin axx cos axx

xsin axcos ax

xn ln xln xxn

x exx ex

x2ex

x2sinx

x2cosx

x2

parts will need to be applied twice this would apply to other similar examples.ex

sinx

cosx

lnxln x1

x(2x + 3)4x(2x + 3)4

x sec2xtanxx sec2xtanx

Trigonometric Identities which could be required:

Which method of integration do you use?

The methods of integrating an expression are:

1) directly writing it down (either by memory or by looking in the formula book);

2) writing the expression in partial fractions;

3) using the method of integration by parts;

4) using a substitution;

5) using a trigonometric identity (such as or )

First look to see if the question tells you (or gives you a hint) about which method of integration you should be using!!

First look to see if the integral is easy to work out: Is it one that you should remember (see my sheet) or is it one in the formula book? Also is the integral easier to work out than it looks e.g.

If you are asked to integrate an algebraic fraction of the form or , you could try writing it in partial fractions.

(Note: you might be asked in the first part of the question to express it in partial fractions)

Examples are:

Other quotients (where the denominator cannot be factorised into linear factors) can usually be integrated using a substitution.

Examples are:

put

put

put

Recognise that the integral is in the form . Examples are:

Use integration by parts to integrate other products, especially when one function is a simple polynomial. Examples are:

To integrate even powers of or , use the formulae or .

Examples are:

To integrate odd powers of or , start by using the formula and then use a substitution.

Examples are:

To integrate sinaxcosbx, use:

sin (A +B) = sinAcosB + sinBcosA

sin (A -B) = sinAcosB - sinBcosA

To integrate sinaxsinbx or cosaxcosbx, use:

cos (A -B) = sinAsinB + cosAcosB

cos (A +B) = sinAsinB - cosAcosB

Integration involving trigonometric identities

Example:

Volumes of revolution: rotating about x axisThe formula is:

Example: The diagram shows the graph of . The region trapped between the lines x = 1, x = 4 and y = 0 is shaded. This region is rotated completely about the x-axis. Find the volume generated.

Volume =

=

=

=

Integration by substitution

Find

Solution: We use the

sin (4x +3x) = sin4xcos3x + sin3xcos4x

sin (4x-3x) = sin4xcos3x sin3xcos4x

Subtracting 1 and 2

sin(4x+3x) sin(4x-3x) = 2sin3xcos4x

So, sin3xcos4x = sin7x sinx

=

=

Example: .

Make the substitution

We get:

This gives:

Example 2: Use the substitution to find .

Solution:

Since ,

So we get

Definite integrals using a substitution

Find .

Use the substitution .

=

x = 0 u = 0

x = 2 u = 4

Take the multipliers outside the integral:

This gives:

But , so .

This expands to give:

Example 1: Find .

This is a suitable candidate for integration by parts with :

Substitute these into the formula:

Example 2: Find .

Here we take :

Substitute these into the formula:

Note: Sometimes it is necessary to use the integration by parts formula twice (e.g. with ).Common examination questions

Example 1: Find .

This can be found using integration by parts if we take .

Substitute these into the formula:

Example 2: Find .

This can be thought of as and so can be integrated by parts with

=

Definite integrals (using by parts)

Example: a) Find the points where the graph of cuts the x and y axes.

b) Sketch the graph of .

c) Find the area of the region between the axes and the graph of .

a) Graph cuts y-axis when x = 0, i.e. at y = 2

Graph cuts the x-axis when y = 0, i.e. when x = 2..

b) The graph looks like:

c) Area is .

So

=

=

Now that weve integrated, we substitute in our limits:

This column gives the calculations for changing the dx to du:

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