Trigonometric Substitution Question

p1akuw47h5r3it

Hi.

I'm currently using MIT's open courseware, and stuck on a problem. The question involves trigonometric substitution. I've posted the question but also linked to the exam paper itself along with the given solution. The question is as follows:

Evaluate *integral sign* dx/(x^2+4)^2 making the substitution x= 2tanu.

I understand why the substitution is 2tanu, because the denominator is of the form sqrt x^2+a^2 = atanu = 2tanu.

My attempt at a solution:

x = 2tanu ; dx = 2sec^2(u)du

sub x and dx into equation:

*integral sign* 2sec^2(u)du/(4tan^2((u)+4)^2

the trig identity sec^2x = tan^2x+1 thus:

*integral sign* 2tan^2(u)+1du/(4tan^2((u)+4)^2

I've squared out the denominator and taken the constants outside:

1/8 *integral sign* tan^2u+1/(tan^2u)(tan^2u)+16

I've taken the 1/16 outside and cancelled off the denominator:

1/16*1/8 *integral sign) 1/tan^2u.

At this point I am stuck and the given solution has a cosine, however I do not understand how they have answered the question. Could I have some help please? Also could someone explain how the limits change.

Cheers

https://ocw.mit.edu/courses/mathemat...s/prexam4a.pdf EXAM

https://ocw.mit.edu/courses/mathemat...rexam4asol.pdf SOLUTIONS

https://ocw.mit.edu/courses/mathemat...rmulasheet.pdf FORMULAE SHEET

TheBody

Hi! Are you sure you have the brackets in the correct position at the start of your post?

None of the links you provided work, so please find the attached to see if it’s what you need.

You made a really good first attempt.

p1akuw47h5r3it

TheBody wrote: »

Hi! Are you sure you have the brackets in the correct position at the start of your post?

None of the links you provided work, so please find the attached to see if it’s what you need.

You made a really good first attempt.

Hi, thankyou for your response and doing out the question. The brackets are in the right place, so it should read as it does. I was unaware the links weren't working. I'm doing exam 4, the practice questions.

I'll give it another go tomorrow, your answer should definitely help so thankyou

https://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/exams/

TheBody

Ok, I can see what you are looking at now. Solution attached (over two pages).

p1akuw47h5r3it

TheBody wrote: »

Ok, I can see what you are looking at now. Solution attached (over two pages).

Brilliant thanks very much for your help. You a mathematician by trade?

TheBody

DanDan6592 wrote: »

Brilliant thanks very much for your help. You a mathematician by trade?

No problem.

Yes, I am a mathematician by trade

p1akuw47h5r3it

TheBody wrote: »

No problem.

Yes, I am a mathematician by trade

Cool, I'm hoping to be myself some day! Applied maths! Cheers again

p1akuw47h5r3it

TheBody,

I'm wondering if you could help me with one more question? It's Q3 from the actual Exam in my previous link.

I've linked my answer. It's incomplete because I've gone wrong. I understand the concept of the questions i.e. with what substitutions to make and why. What has got me on this question is the my use of indices. I'm aware indices are quite basic but it's been a while!!

In the question the denominator is given as (4+x^3)^3/3 which I have changed to the power of 1.

In particular, in the given answer the denominator goes from (4+ 4tan^2 u)^3/2 to (8sec^3 u). I'm unsure as to exactly this occurs although I am aware of the relationship sec^x = tan^2 x +1. I just cannot make sense of it exactly.

Cheers.

TheBody

I think there is a typo in the question. It's not the same as the problem they solve in the solutions.

The problem they do in the solutions is standard enough.