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    (Original post by newblood)
    if youre referring to the original integral, i think he also changed the signs as well to change the limits, just for ease of manipulation id imagine
    Changing order of limits is often not justified though.
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    (Original post by james22)
    Changing order of limits is often not justified though.
    i havent done uni maths so i'll step away here

    I know this isnt a maths help thread and i dont wish to derail it but since people are currently online and no-one has posted solutions recently : can anyone provide some clarity on this thread;

    http://www.thestudentroom.co.uk/show....php?t=2600276

    i can not understand why its not applying to differentiation/calculus - is that in fact true?
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    (Original post by james22)
    Why are you justified in swapping the order of limits here?
    DCT of course.

    (use \delta_{n,k}=a_{n,2k-1}-a_{n,2k}<k^{-2} where a_{n,m}=n\big(mn-1\big)^{-1} for instance)
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    If I drop a cannon ball off the Eiffel Tower, where does it land?
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    (Original post by cheeseman552)
    If I drop a cannon ball off the Eiffel Tower, where does it land?
    On the ground.
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    (Original post by cheeseman552)
    If I drop a cannon ball off the Eiffel Tower, where does it land?
    By the way, a really really neat proof that speed of falling is independent of mass is:
    Drop two identical cannonballs next to each other. They will obviously fall at the same speed as each other.
    Repeat the experiment but this time tie the cannonballs together with a loose piece of string.
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    (Original post by cheeseman552)
    If I drop a cannon ball off the Eiffel Tower, where does it land?
    It doesn't... O_o

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    In how many different ways can you find

    \displaystyle\int e^x \cos(x) \cosh(x) dx

    Try doing it using only the cyclic nature of teh derivatives of the functions in the integrand.
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    (Original post by james22)
    In how many different ways can you find

    \displaystyle\int e^x \cos(x) \cosh(x) dx

    Try doing it using only the cyclic nature of teh derivatives of the functions in the integrand.
    

\displaystyle\quad \int e^x\cos{x}\cosh{x}\,\mathrm{d}x\  \

\displaystyle=\Re\:\frac{1}{2} \int e^{(2+i)x}+e^{ix}\,\mathrm{d}x\\

\displaystyle=\Re\:\frac{e^{(2+i  )x}}{2(2+i)}+\frac{e^{ix}}{2i}+C  \\

\displaystyle=\Re\:\frac{(2-i)e^{(2+i)x}}{10}-\frac{ie^{ix}}{2}+C\\

\displaystyle=\frac{\sin x}{2}+\frac{e^{2x}\cos{x}}{5}+ \frac{e^{2x}\sin{x}}{10}+C
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    (Original post by james22)
    In how many different ways can you find

    \displaystyle\int e^x \cos(x) \cosh(x) dx

    Try doing it using only the cyclic nature of teh derivatives of the functions in the integrand.
    There's a host of different IBP to do, although ultimately you end up doing the same things whichever parts you take.
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    (Original post by Indeterminate)
    Problem 458***

    For u, v >0 show that

    \displaystyle \int^{\infty}_{-\infty} \dfrac{u\cos vx - x\sin vx}{(u^2+x^2)^2} dx = \dfrac{\pi}{2u^2e^{uv}}.
    This is trivialish via complex analysis, just use de Moivre and a semicircular contour.
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    Problem 461 **/***

    Evaluate

    \displaystyle \int \dfrac{\log(1+x)}{x^2 + 1} \text{ d}x
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    (Original post by Khallil)
    Problem 461 **/***
    Presumably over [0,1]. In any case, this has been posted already (and something very similar to 458 was posted as well). I think the thread needs a break from integration
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    (Original post by Lord of the Flies)
    Presumably over [0,1]. In any case, this has been posted already (and something very similar to 458 was posted as well). I think the thread needs a break from integration
    I had a go at the indefinite integration but kept going around in circles. All of my substitutions kept spiraling me around to the same form

    Also, I've got a quick question. For integrals with lower and upper bounds, say \displaystyle \int_{a}^{b}, are we integrating over the interval [a, b] or (a, b) ?
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    (Original post by Khallil)
    Problem 461 **/***

    Evaluate

    \displaystyle \int \dfrac{\log(1+x)}{x^2 + 1} \text{ d}x
    Love that problem, though I'm sure I've seen it on this thread before? At any rate I'll leave it for someone who's not done it before
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    Problem 462 ***

    A is a symmetric matrix with eigenvectors u and v and with corresponding, distinct eigenvalues λ, μ. Show that the eigenvectors are orthogonal.

    (Sorry if this question has been posted before, I don't want to look through 461 other questions :L, also I'm sorry if it's too easy/too hard/no one has any idea what I'm on about)
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    (Original post by Flauta)
    Love that problem, though I'm sure I've seen it on this thread before? At any rate I'll leave it for someone who's not done it before
    The indefinite integral? :eek:
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    (Original post by Khallil)
    I had a go at the indefinite integration but kept going around in circles. All of my substitutions kept spiraling me around to the same form
    Ah, definitely no closed form for this one.

    Also, I've got a quick question. For integrals with lower and upper bounds, say \displaystyle \int_{a}^{b}, are we integrating over the interval [a, b] or (a, b) ?
    The Riemann integral is over [a,b] (in the sense that it requires f to be defined everywhere in the interval of integration). Improper integrals are then defined as limits.
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    (Original post by Khallil)
    The indefinite integral? :eek:
    No of course not But the definite integral from 0 to 1 only needs a substitution to evaluate

    (Will admit that I misread the question and thought it said the definite integral from 0 to 1)
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    (Original post by Khallil)
    I had a go at the indefinite integration but kept going around in circles. All of my substitutions kept spiraling me around to the same form

    Also, I've got a quick question. For integrals with lower and upper bounds, say \displaystyle \int_{a}^{b}, are we integrating over the interval [a, b] or (a, b) ?
    Notice that for nice enough integrands, they're the same, because individual points as discontinuities don't alter the value of the integral.

    (Original post by rayquaza17)
    Problem 462 ***

    A is a symmetric matrix with eigenvectors u and v and with corresponding, distinct eigenvalues λ, μ. Show that the eigenvectors are orthogonal.

    (Sorry if this question has been posted before, I don't want to look through 461 other questions :L, also I'm sorry if it's too easy/too hard/no one has any idea what I'm on about)
    Yay
    Spoiler:
    Show
    Au = \lambda u, Av = \mu v, so v^T (Au) = \lambda v^T u by premultiplying the first equation by v^T; also v^T A u = (A v)^T u by symmetry, and that is \mu v^T u. Hence \lambda v^T u = \mu v^T u; but \lambda, \mu are not the same, so v^T u = 0.
 
 
 
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