MYTUTOR SUBJECT ANSWERS

375 views

How and when do you use integration by parts?

Integration by parts is a method of integration used when you are attempting to integrate a function which is the product of two functions. If the two products can be expanded there is usually an easier way to integrate them than integration by parts. For example, x2(x - 4) is easier to integrate when expanded to x3 - 4x2.

The general form of the equation for integration by parts is:

∫f(x)g’(x)dx = f(x)g(x) - ∫g(x)f’(x)dx

where f’(x) is the derivative of f(x). It is also commonly seen as:

∫u dv/dx dx = uv - ∫v du/dx dx

where u and v are both function of x.

A good guideline when deciding which function to use as u (or f(x)) is the acronym LIATE:

Logarithmic e.g. ln(x)

Inverse trigonometry e.g. sin-1(x)

Algebraic e.g. x

Trigonometry e.g. sin(x)

Exponential e.g. ex.

Step 1:

Split the integrand (function to be integrated) in to its 2 products.

E.g. ∫xln(x)dx can be split in to x and ln(x).

Step 2:

Decide which function should be u and which should be dv/dx.

E.g. x is algebraic, ln is logarithmic. Logarithmic comes before algebraic in LIATE so u = ln(x) and dv/dx = x.

Step 3:

Find du/dx and v by differentiating and integrating u and dv/dx respectively.

E.g. u = ln(x), du/dx = x-1, dv/dx = x and v = x2/2

Step 4:

Substitute the variables in to the equation for integration by parts.

E.g. ∫xln(x)dx = ln(x)x2/2 - ∫x-1x2/2 dx = ln(x)x2/2 - ∫x/2 dx.

Step 5:

Evaluate the new integral.

E.g ∫xln(x)dx = ln(x)x2/2 - x2/4 + c = x2/4 (2ln(x) - 1) + c where c is a constant of integration.

Step 5 may require you to perform integration by parts again. Also LIATE does not work in every situation. If it does not work, switch the products used for u and dv/dx and try again. 

Ashley P. A Level Physics tutor, A Level Maths tutor, A Level Further...

1 year ago

Answered by Ashley, an A Level Maths tutor with MyTutor


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

176 SUBJECT SPECIALISTS

£20 /hr

Larry R.

Degree: Mathematics (Bachelors) - Warwick University

Subjects offered: Maths, Further Mathematics

Maths
Further Mathematics

“I am a Mathematics student at Warwick University. My enthusiasm for all things mathematical has no bounds and it's not just Maths that interests me but also the way that people learn it.  I am patient and friendly. Over the last few y...”

£22 /hr

Kirsty S.

Degree: Mathematics (Bachelors) - Warwick University

Subjects offered: Maths, Spanish+ 1 more

Maths
Spanish
Further Mathematics

“Hello, I'm Kirsty and I am here to help you with Maths. Maths will soon become the exam that you will look forward to, so you can show off how much you know! Before I begin any tuition I find out exactly what you would like to get ou...”

£20 /hr

Noam T.

Degree: Mathematics with Mathematical Physics (Bachelors) - University College London University

Subjects offered: Maths, Physics

Maths
Physics

“About Myself: I am an undergraduate student at UCL studying Mathematics with Mathematical Physics. Hoping to go into research after my degree, I have a true passion for my subjects and my aim is to pass on that passion to students in ...”

About the author

Ashley P.

Currently unavailable: for regular students

Degree: Physics (Masters) - Southampton University

Subjects offered: Maths, Physics+ 1 more

Maths
Physics
Further Mathematics

“Hi! I'm a third year physics student with a keen interest in science and maths looking to inspire others.”

You may also like...

Other A Level Maths questions

How do I add up the integers from 1 to 1000 without going insane?

How do you differentiate parametric equations?

How can I remember the difference between differentiation and integration?

How to gain an inverse function

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