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Core 3: Integration Skills


1)           Key integration facts

2)          Indefinite integration by substitution

3)          Indefinite Integration by-parts

4)          Key trigonometry facts

5)          Definite integration by by-parts

6)          Definite integration by substitution and some extra facts


Key integration facts

Make sure you learn and comfortable with the following facts


Integration by substitution

Accustom yourself with the following integral types

This transforms to



The terms outside the bracket is the same order as the differential of the term inside the bracket.

Let u=

Observe how the  ‘cancels’



When the power of x is at most 1 both outside and inside the bracket

Let u=3x+2

Rearrange to find x in terms of u.

Substitute this in for x

Note at the end that




The numerator is the same order as the differential of the denominator.


Observe how the x’s cancel out


This one is quite difficult


Re-arrange to get


Much easier


The  cancel


Indefinite Integration by-parts

Work through these examples carefully, it is actually a very easy technique once the technical notation has been resolved.

There are actually only a few examples which can be asked on an A-level example.
The general formula is:
The hard part is to know how to set up the actual formula.


Generally, let  be the term which will significantly simplify the most when differentiated.


Therefore we define:

Remember to do the final integration. Students often forget this.




Where one needs to be careful is when one meets the integral of the natural logarithm.

 substantially simplifies when differentiated so:



The trick is to write


 substantially simplifies when differentiated so:



Key integration facts

Even with modern calculator able work out exact trigonometry expressions in terms of surds, it is nevertheless good practice to understand and be able to use a number of key trigonometry values. This is especially the case if you aim to apply to study at university. University examinations and entrance examinations often rely up this knowledge!

Principle angle facts:


















Using symmetry, so you can use these to work out the exact value of a number of expressions between 0 and 2π.


Definite integration by-parts

We integrated this earlier on.

Remember that the +c is not required for a definite integrand.

We integrated this earlier.

Also not that



Definite integration by substitution


See if you can work out an answer for this?

We integrated this earlier on.

The key idea here is to transform the limits using the transformation .

An extra fact worth knowing

Particular example

General case