MYTUTOR SUBJECT ANSWERS

259 views

What is the integral of x sin(x) dx?

Find the following integral: ∫ x sin(x) dx

This question is a good candidate for the integration by parts method, as it is the product of two different 'parts'.

Recall that if you have an integral of the form

∫ u(dv/dx) dx

it can be written as

uv – ∫ v(du/dx) dx.

We need to decide which part we will differentiate (as in, which part is u), and which part we will integrate (as in, which part is dv/dx). 

We can note that continuously differentiating sin(x) results in a loop of cos(x), –sin(x), –cos(x), sin(x)..., whereas differentiating x once gives 1.

From this, it seems to make sense that we would want to differentiate the x part (so u is x) and therefore integrate the sin(x) part (so dv/dx is sin(x) ). So, let

u = x, which implies du/dx = 1

and let

dv/dx = sin(x). Integrating this to get v gives v = –cos(x).

So our integral is now of the form required for integration by parts.

∫ x sin(x) dx 

=  ∫ u(dv/dx) dx

= uv –  ∫ v(du/dx) dx

= –x cos(x) –  ∫ –cos(x)*1 dx

= –x cos(x) –  ∫ –cos(x) dx

= –x cos(x) +  ∫ cos(x) dx

The integral of cos(x) is equal to sin(x). We can check this by differentiating sin(x), which does indeed give cos(x). Finally, as with all integration without limits, there must be a constant added, which I'll call c. So the final answer is 

∫ x sin(x) dx = –x cos(x) +  sin(x) + c

Shaun F. GCSE Maths tutor, A Level Maths tutor

11 months ago

Answered by Shaun, an A Level Maths tutor with MyTutor

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

184 SUBJECT SPECIALISTS

£20 /hr

Mrinank S.

Degree: Engineering (Masters) - Cambridge University

Subjects offered: Maths, Physics+ 5 more

Maths
Physics
Further Mathematics
Economics
Chemistry
.PAT.
-Oxbridge Preparation-

“Hey there! I'm a second year undergraduate student at the University of Cambridge, studying Engineering. I am really passionate about using science and maths to make the world around us a better place.  I am a patient and friendly tu...”

£20 /hr

Tiarnan B.

Degree: MChem (Masters) - York University

Subjects offered: Maths, Geography+ 2 more

Maths
Geography
Chemistry
-Personal Statements-

“About Me: I am a student of Chemistry at the University of York studying for a masters degree and am a self-confessed science nerd! My goal is to give you the help you need, build your confidence and push you to be the best you can. A...”

£20 /hr

Raghav A.

Degree: Chemical Engineering (Masters) - Bath University

Subjects offered: Maths, Physics+ 2 more

Maths
Physics
Chemistry
-Personal Statements-

“About Me I am currently studying Chemical Engineering at the University of Bath. I have always had a real passion for mathematics and science and hope that my tutorials will instill that in you, too. I am very patient and friendly. I...”

MyTutor guarantee

About the author

£20 /hr

Shaun F.

Degree: Mathematics with International Study (Masters) - Exeter University

Subjects offered: Maths

Maths

“Hi, I'm Shaun! I'm currently studying for a masters in Mathematics at the University of Exeter..”

MyTutor guarantee

You may also like...

Other A Level Maths questions

How do I integrate cos^2(x)?

How do I differentiate y = (3 + 6x)^5 with respect to x

What are the roots of 3x^2 + 13x + 4 ?

Please explain Pythgoras Theorem

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