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    A couple of (A-level) integrals for you guys to enjoy!

    Problem 1


    Find

    \displaystyle \int \dfrac{\sin^3\left(\frac{x}{2} \right)}{\cos\left(\frac{x}{2} \right)\sqrt{\cos^3 x + \cos^2 x + \cos x}} \ dx



    Problem 2


    Find

    \displaystyle \int \dfrac{\ dx}{\sqrt{1+x} + \sqrt{1-x} + 2}

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    (Original post by Indeterminate)
    A couple of (A-level) integrals for you guys to enjoy!

    Problem 1


    Find

    \displaystyle \int \dfrac{\sin^3\left(\frac{x}{2} \right)}{\cos\left(\frac{x}{2} \right)\sqrt{\cos^3 x + \cos^2 x + \cos x}} \ dx



    Problem 2


    Find

    \displaystyle \int \dfrac{\ dx}{\sqrt{1+x} + \sqrt{1-x} + 2}

    I approve
    shame I am teaching
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    Disappointed by the fact that we haven't had any responses yet :cry2:
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    (Original post by Indeterminate)
    You guys disappoint me :cry2:
    I worked on the first one for literally an hour and couldn't get anywhere further than all multiples/orders of cosxs :cry:
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    (Original post by Indeterminate)
    Disappointed by the fact that we haven't had any responses yet :cry2:
    My integration's a bit crap. Something like  x = \cos 2u on the 2nd?
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    (Original post by Student403)
    I worked on the first one for literally an hour and couldn't get anywhere further than all multiples/orders of cosxs :cry:
    :console:

    Getting rid of the half angles is the key to this one. How good is your trig?

    (Original post by 16Characters....)
    My integration's a bit crap. Something like  x = \cos 2u on the 2nd?
    Try it by all means
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    (Original post by Indeterminate)
    :console:

    Getting rid of the half angles is the key to this one. How good is your trig?



    Try it by all means
    I got rid of them all and got down to a large expression just involving a bunch of cosx's multiplied by each other and a few square roots and squares :/
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    (Original post by Student403)
    I got rid of them all and got down to a large expression just involving a bunch of cosx's multiplied by each other and a few square roots and squares :/
    I see. Well I'm sure Zacken will have a few ideas.

    The purpose of this thread is to get you guys talking Think of me as a last resort
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    (Original post by 16Characters....)
    My integration's a bit crap. Something like  x = \cos 2u on the 2nd?
    (Original post by Indeterminate)
    I see. Well I'm sure Zacken will have a few ideas.

    The purpose of this thread is to get you guys talking Think of me as a last resort
    I'm thinking u = \sqrt{1+x}.
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    Re my last post: nopes.
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    (Original post by Zacken)
    I'm thinking u = \sqrt{1+x}.
    You've both made interesting suggestions However, it's not a straightforward, single-substitution integral.

    I'd go with 16Characters....myself
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    Fairly sure I can see how to get the second one out. I'll give it a go soon.

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    (Original post by Indeterminate)
    You've both made interesting suggestions However, it's not a straightforward, single-substitution integral.

    I'd go with 16Characters....myself
    Yeah, I realised. - you should totally give difficulty ratings for your integrals. (*, **, *** perhaps?).
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    (Original post by Zacken)
    Yeah, I realised. - you should totally give difficulty ratings for your integrals. (*, **, *** perhaps?).
    Haha, I'd say that they're both **

    I could do worse
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    I'm down to: \displaystyle \int \frac{-\sin 2x}{\sin \left(x + \frac{\pi}{4}\right) + 1} \, \mathrm{d}x
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    (Original post by Zacken)
    I'm down to: \displaystyle \int \frac{-\sin 2x}{\sin \left(x + \frac{\pi}{4}\right) + 1} \, \mathrm{d}x
    So far, so good. Time to sort out the trig

    Hint:
    Spoiler:
    Show
    \sin \left(x + \dfrac{\pi}{4}\right) = ....
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    (Original post by Indeterminate)
    So far, so good. Time to sort out the trig

    Hint:
    Spoiler:
    Show
    \sin \left(x + \dfrac{\pi}{4}\right) = ....
    ... :cry2: But I moved from \cdots to the collected sine term.

    I'm on \displaystyle \int \frac{\cos 2x}{\sin x+1} \, \mathrm{d}x right now.
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    (Original post by Zacken)
    ... :cry2: But I moved from \cdots to the collected sine term.

    I'm on \displaystyle \int \frac{\cos 2x}{\sin x+1} \, \mathrm{d}x right now.
    Carry on. Let's see what you get!
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    (Original post by Indeterminate)
    A couple of (A-level) integrals for you guys to enjoy
    Hi, these integrals are beyond me!
    I just wanted to say that I think the title of this thread is amazing! I don't know if you did it on purpose or not (you probably did) but it's so clever the qay it's like A tale of two cities

    Posted from TSR Mobile
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    (Original post by Matrix123)
    Hi, these integrals are beyond me!
    I just wanted to say that I think the title of this thread is amazing! I don't know if you did it on purpose or not (you probably did) but it's so clever the qay it's like A tale of two cities

    Posted from TSR Mobile
    Yup I'm quite a fan of Dickens and A Tale of Two Cities is one of my most favourite works of his :awesome:
 
 
 
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