Turn on thread page Beta
    • Thread Starter
    Offline

    6
    ReputationRep:
    Name:  Screenshot_2016-03-13-11-55-37.png
Views: 144
Size:  131.3 KB
    I am not sure why the integral of sec^(2)xtan^(2)x is 1/3tan^(3)x.
    Please can anyone show me the steps to get there. Much appreciated!
    Offline

    18
    ReputationRep:
    (Original post by QueenOfNachos)
    Name:  Screenshot_2016-03-13-11-55-37.png
Views: 144
Size:  131.3 KB
    I am not sure why the integral of sec^(2)xtan^(2)x is 1/3tan^(3)x.
    Please can anyone show me the steps to get there. Much appreciated!
    Well, think of a substitution.
    • Thread Starter
    Offline

    6
    ReputationRep:
    (Original post by zetamcfc)
    Well, think of a substitution.
    Ah yes. How did I not think of that. Got it now, thanks a lot!!
    Offline

    11
    ReputationRep:
    (Original post by QueenOfNachos)
    Name:  Screenshot_2016-03-13-11-55-37.png
Views: 144
Size:  131.3 KB
    I am not sure why the integral of sec^(2)xtan^(2)x is 1/3tan^(3)x.
    Please can anyone show me the steps to get there. Much appreciated!
    If you start with y=\tan^3 x and write u=\tan x \Rightarrow y=u^3 then by the chain rule:

    \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx} = 3u^2 \times \sec^2 x.

    Can you finish this?
    • Thread Starter
    Offline

    6
    ReputationRep:
    (Original post by atsruser)
    If you start with y=\tan^3 x and write u=\tan x \Rightarrow y=u^3 then by the chain rule:

    \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx} = 3u^2 \times \sec^2 x.

    Can you finish this?
    Yes I can, thank you very much!
    Offline

    12
    ReputationRep:
    (Original post by QueenOfNachos)
    Name:  Screenshot_2016-03-13-11-55-37.png
Views: 144
Size:  131.3 KB
    I am not sure why the integral of sec^(2)xtan^(2)x is 1/3tan^(3)x.
    Please can anyone show me the steps to get there. Much appreciated!
    I'm a bit late to the party, but I prefer to do these integrals by recognition (also referred to as "reverse chain rule" rather than substitution - it's a little quicker than using a sub. I'll show you what I mean.

    We know that \displaystyle \frac{\mathrm{d} }{\mathrm{d} x}\left ( \tan^{3}x \right ) = 3\sec^{2}x\tan^{2}x via the chain rule, right? Since we know the derivative of this, we can say that
    \displaystyle \Rightarrow \int 3 \sec^{2}x\tan^{2}x \ \mathrm{d}x = \tan^{3}x + \mathrm{C_{1}}
    Won't be too tough finding the integral from here, I hope!
    Spoiler:
    Show
    \displaystyle \boxed{\therefore \int \sec^{2}\tan^{2}x \ \mathrm{d}x = \frac{\tan^{3}x}{3} + \mathrm{C}}
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: March 13, 2016
The home of Results and Clearing

3,010

people online now

1,567,000

students helped last year

University open days

  1. Bournemouth University
    Clearing Open Day Undergraduate
    Wed, 22 Aug '18
  2. University of Buckingham
    Postgraduate Open Evening Postgraduate
    Thu, 23 Aug '18
  3. University of Glasgow
    All Subjects Undergraduate
    Tue, 28 Aug '18
Poll
How are you feeling about GCSE results day?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.