LEGACY Integration

A-Level Pure

Common Results

Reverse Chain Rule

Substitution

Sometimes you can simplify an integral by changing the variable. This process is similar to using the chain rule in differentiation and is called integration by substitution.

Integration By Parts

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A-Level Further Core 2

Improper integrals

The integral is improper if:

• One or both of the limits are or
• is undefined at any point in

To find , consider and compute the limit (if it converges) as For an integral , split into 2 integrals: ,

Mean Value

The mean value of a function in the interval is:

This is because the integral is taking infinite samples. If has mean value over the interval then:

• has mean value
• has mean value ,

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A-Level Further Pure 2

Reduction Formula

The Reduction formula allows you to write an integral as a recurrence relation. This is generally used for integrals with high powers that would require many integration by parts iterations.

You can use the reduction formula in conjunction with a substitution and the Method of Differences to compute tricky summations.

Arc Length

Cartesian

The arc length of from and is:

Parametric

For , , the arc length from to is:

Polar

For , the arc length from half lines to is:

Surface Area of Volume of Revolution

The area, , of the surface generated when the arc on the curve is rotated completely about the -axis is , and about the -axis is .

Cartesian

About the -axis:

About the -axis:

Parametric

About the -axis:

About the -axis:

Polar

About the initial line

About the line

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