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An A-level integral Watch

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    I have decided to take a leaf out of TeeEm's book

    Evaluate

    \displaystyle \int e^{x\sin x + \cos x} \left(\dfrac{x^4 \cos^3 x - x \sin x + \cos x}{x^2 \cos^2 x}\right) \ dx

    I initially posted this on 'the hard integral thread', but I think it deserves a wider audience because, after all, it only requires knowledge of the regular A-level Maths syllabus :yep:

    Enjoy!
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    (Original post by Indeterminate)
    I have decided to take a leaf out of TeeEm's book in order to invoke some kind of a discussion on this forum

    Evaluate

    \displaystyle \int e^{x\sin x + \cos x} \left(\dfrac{x^4 \cos^3 x - x \sin x + \cos x}{x^2 \cos^2 x}\right) \ dx

    I initially posted this on 'the hard integral thread', but I think it deserves a wider audience because, after all, it only requires knowledge of the regular A-level Maths syllabus :yep:

    Enjoy!
    very good
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    (Original post by TeeEm)
    very good
    Shame we haven't had any replies yet
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    (Original post by Indeterminate)
    Shame we haven't had any replies yet
    maths died about a year ago ...
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    (Original post by Indeterminate)
    Shame we haven't had any replies yet
    I'd give it a bash but I'm swamped with FP3.
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    (Original post by Zacken)
    I'd give it a bash but I'm swamped with FP3.
    Lol, that's fair enough

    So I think it's time to tag a few people...
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    I'll give it a try, be back soonish with my answer hopefully
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    (Original post by Zacken)
    I'd give it a bash but I'm swamped with FP3.
    I'd give it a bash but I can't be bothered as it looks hard.

    Though maybe you can look for an IBP inspired approach? The exponential would seem to point that way.
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    (Original post by Indeterminate)
    Lol, that's fair enough

    So I think it's time to tag a few people...
    I'm honoured to be tagged in one of Indeterminate's integral posts, but my integration skills are quite laughable.

    I've attempted to separate them into individual functions using simplification and have tried trigonometric substitutions, but it's not looking too pretty... Any hints?
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    (Original post by aymanzayedmannan)
    I'm honoured to be tagged in one of Indeterminate's integral posts, but my integration skills are quite laughable.

    I've attempted to separate them into individual functions using simplification and have tried trigonometric substitutions, but it's not looking too pretty... Any hints?
    You don't need to use a substitution. Play around with what's inside the brackets and see what you get
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    (Original post by drandy76)
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    Try differentiating that and then compare it to the expression to be integrated


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    (Original post by KFazza)
    Try differentiating that and then compare it to the expression to be integrated


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    oh dear
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    I don't think this can be classed as a solution cause I did a lot of this question by observation which I tried to explain with the notes as I go along


    IMPORTANT EDIT: at the end there should be a minus not a plus. IntA - IntB

    But anyway is this correct? Hopefully you can follow along..
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    Having had a quick look through all the responses, I am now delighted to announce that the winners are Student403 and KFazza

    I guess I should write up a solution using LATEX, so here goes...
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    Let I denote our integral.

    We have

    I = \displaystyle \int e^{x\sin x + \cos x} \left( x^2 \cos x + \dfrac{1}{x^2 \cos x} - \dfrac{\sin x}{x \cos^2 x}\right) \ dx

     = \displaystyle \int e^{x\sin x + \cos x} \left( x^2 \cos x + 1 + \dfrac{1}{x^2 \cos x} - \dfrac{\sin x}{x \cos^2 x} - 1 \right) \ dx

     = \displaystyle \int e^{x\sin x + \cos x} \left[\left(\dfrac{1}{x^2 \cos x} - \dfrac{\sin x}{x \cos^2 x} + 1\right) + x\cos x \left(x - \dfrac{1}{x \cos x}\right)\right] \ dx

     = \displaystyle e^{x \sin x + \cos x} \left( x- \dfrac{1}{x\cos x}\right) + C

    by recognition

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    (Original post by Indeterminate)
    Having had a quick look through all the responses, I am now delighted to announce that the winners are Student403 and KFazza

    I guess I should write up a solution using LATEX, so here goes...
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    Let I denote our integral.

    We have

    I = \displaystyle \int e^{x\sin x + \cos x} \left( x^2 \cos x + \dfrac{1}{x^2 \cos x} - \dfrac{\sin x}{x \cos^2 x}\right) \ dx

     = \displaystyle \int e^{x\sin x + \cos x} \left( x^2 \cos x + 1 + \dfrac{1}{x^2 \cos x} - \dfrac{\sin x}{x \cos^2 x} - 1 \right) \ dx

     = \displaystyle \int e^{x\sin x + \cos x} \left[\left(\dfrac{1}{x^2 \cos x} - \dfrac{\sin x}{x \cos^2 x} + 1\right) + x\cos x \left(x - \dfrac{1}{x \cos x}\right)\right] \ dx

     = \displaystyle e^{x \sin x + \cos x} \left( x- \dfrac{1}{x\cos x}\right) + C

    by recognition

    Thanks I cannot understand how people can write in LaTeX :O
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    (Original post by Student403)
    Thanks I cannot understand how people can write in LaTeX :O
    Congrats. :woo:
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    (Original post by Zacken)
    Congrats. :woo:
    Thanks also to you b/c the Cambridge thread won a community award!
 
 
 
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