What methods are there for integration?

There are five methods that can help you integrate a function. All these methods do is simplify the expression you're trying to integrate until you are left with something that you can recall what it integrates to.

1. Trigonometric Identities

This is where you have an expression made up of trigonometric functions (i.e. sin, cos, tan, sec, cosec, cot). You can use one or more of the trig identities, that you should be able to recall from memor,y to rewrite the expression into something that you can integrate with more ease.

Identities you'll need to memorise:


sin2(x)+cos2(x)=1 (here you can derive identies involving cot, sec and cosec by diving by sin or cos)





These are the simplest forms of trig identities that you'll need to know. Using them will make it a lot easier to integrate a tricky expression.

2. Partial Fractions

The method of partial fractions can be used when you have to integrate a fraction where the denominator is a product of two different functions. 

i.e  (a+b)/cd ,

you begin with supposing that

(a+b)/cd = e/c + f/d (then multiply by cd)

a + b = ed +fc (at this stage you should be able to solve for e and f)

It may then be easier to integrate this simpler expression of e/c + f/d.

3. Substituon

This method is probably the most common and the best to try if you are still stuck after using trig identities and/or partial fractions.

This involves picking a part of the function say x2 and letting it equal an arbitary letter, for example u. You then differentiate this picked out part:       u = x2 ,  du/dx =2x , thus dx= 1/2x du

You can then substitute this 1/2x du for dx in the original expression. This should hopefully simplify the expression and make it easy to integrate with respect to u.

Knowing what substitution to use in different questions comes from experience, so do a lot of practice and you'll get the hang of it.

4. By Parts

This method is derived from the product rule however it is easiest to just memorise it in this form:

∫ u(dv/dx) dx = uv − ∫ v(du/dx) dx .

where u and v are any functions.

5. Inspection

This may be the trickiest method but there will be cases where it is the only one you can use.

This involves trying to 'guess' what functions could be differentiated to get your original expression.

With practice this becomes easier.

Integration can be really tricky but with a lot practice you can begin to spot the best ways to simplify expressions into formats that you know the integral of.

Celine A. GCSE Maths tutor, A Level Maths tutor, GCSE Further Mathema...

2 months ago

Answered by Celine, an A Level Maths tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


PremiumDaniel R. A Level Maths tutor, A Level Further Mathematics  tutor, GC...
View profile
£24 /hr

Daniel R.

Degree: Mathematics (Bachelors) - Durham University

Subjects offered: Maths, Physics+ 1 more

Further Mathematics

“About me I’m a first year student studying Maths with European Studies at Durham. I have recently taken my A levels, achieving A* in Maths and Further Maths, so I am familiar with the course content and what the examiners are looking ...”

Guy P. GCSE Maths tutor, A Level Maths tutor, A Level Further Mathema...
View profile
£20 /hr

Guy P.

Degree: Mathematics (Masters) - Warwick University

Subjects offered: Maths, Further Mathematics + 2 more

Further Mathematics

“About:Hi. I am a 2nd Year Mathematics student at the University of Warwick. I achieved a comfortable First in Year 1 and have continued this trend into my second year. Even from an early age, I have had a burning passion to engage m...”

Alex D. IB Maths tutor, 11 Plus Maths tutor, GCSE Maths tutor, A Leve...
View profile
£20 /hr

Alex D.

Degree: Chemical Engineering (Masters) - Edinburgh University

Subjects offered: Maths, Physics+ 1 more


“I have a lot of experience working with children as a young scout leader volunteering at the cubs for a year and a half as well as being a mentor to a primary 4 and a primary 7 class for a year. I also have a PVG Disclosure Scotland c...”

MyTutor guarantee

About the author

Celine A. GCSE Maths tutor, A Level Maths tutor, GCSE Further Mathema...
View profile
£20 /hr

Celine A.

Degree: Mathematics (Bachelors) - Bristol University

Subjects offered: Maths, Further Mathematics

Further Mathematics

“Hello! I'm Celine and I am currently in my second year of my Maths degree at the University of Bristol.  I offer any level of Maths tutoring up to and including Maths and Further Maths A-level.  We can go through your course and home...”

MyTutor guarantee

You may also like...

Other A Level Maths questions

Given that d/dx(cosx)=-sinx show that d/dx(secx)=secx(tanx)

Find the derivative of x^x

What qualifications and experience do you have at this level?

What is Taylor Series

View A Level Maths tutors


We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss