MYTUTOR SUBJECT ANSWERS

126 views

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:

tan(x)=sin(x)/cos(x)

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

sin(a+b)=sin(a)cos(b)+sin(b)cos(a) 

sin(a-b)=sin(a)cos(b)-sin(b)cos(a) 

cos(a+b)=cos(a)cos(b)-sin(b)sin(a)

cos(a-b)=cos(a)cos(b)+sin(b)sin(a)

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...

3 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

175 SUBJECT SPECIALISTS

£20 /hr

Liam B.

Degree: Mathematics & Economics (Joint Honours) (Bachelors) - Durham University

Subjects offered: Maths, Economics

Maths
Economics

“About Me: I am currently an undergraduate student studying Mathematics & Economics at Durham University. I have always found myself enjoying Maths during my time learning, and I hope to be able to share this joy with yourself by ensuri...”

£20 /hr

Munroop P.

Degree: Economics (Bachelors) - Warwick University

Subjects offered: Maths, Spanish+ 1 more

Maths
Spanish
Economics

“I am an Economics student at Warwick University. I am passionate about Economics but I also love Maths and Spanish, and so I am thrilled to be able to take Maths and Spanish modules within my Economics degree course.  I have experienc...”

£20 /hr

Seb G.

Degree: Mathematics (Masters) - Bath University

Subjects offered: Maths

Maths

“Me: I'm a second year maths student at the University of Bath. I fell in love with maths after a rocky start with the subject, so believe that I can help people regardless of whether they like maths or not.  I've been tutoring maths l...”

MyTutor guarantee

About the author

£20 /hr

Celine A.

Degree: Mathematics (Bachelors) - Bristol University

Subjects offered: Maths, Further Mathematics

Maths
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

What is the probability that a leap year has 53 Sundays?

How do I solve equations like 3sin^2(x) - 2cos(x) = 2

Why do I have to add +c when I integrate?

How do I solve this inequality: x^2>2x ?

View A Level Maths tutors

Cookies:

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

mtw:mercury1:status:ok