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Math Help - Intergrals of Inverse Trig Functions

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    Senior Member polymerase's Avatar
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    Intergrals of Inverse Trig Functions

    I've actually just noticed as I was flipping through my calculus textbooks that none of the books I have actually mention or show or listed in the table of integrals how to get these types of integrals: \int sin^{-1}\;x or \int cos^{-1}\;x etc. Can anyone show me how to actually calculate these types.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by polymerase View Post
    I've actually just noticed as I was flipping through my calculus textbooks that none of the books I have actually mention or show or listed in the table of integrals how to get these types of integrals: \int sin^{-1}\;x or \int cos^{-1}\;x etc. Can anyone show me how to actually calculate these types.
    integration by parts should work.

    1 is the function you integrate, the inverse trig function is the one you differentiate
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by polymerase View Post
    I've actually just noticed as I was flipping through my calculus textbooks that none of the books I have actually mention or show or listed in the table of integrals how to get these types of integrals: \int sin^{-1}\;x or \int cos^{-1}\;x etc. Can anyone show me how to actually calculate these types.
    Do it by parts. For example
    \int sin^{-1}(x)~dx = x~sin^{-1}(x) - \int \frac{x}{\sqrt{1 - x^2}}~dx

    -Dan

    Edit: Is there an echo in here?
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    Quote Originally Posted by polymerase View Post
    I've actually just noticed as I was flipping through my calculus textbooks that none of the books I have actually mention or show or listed in the table of integrals how to get these types of integrals: \int sin^{-1}\;x or \int cos^{-1}\;x etc. Can anyone show me how to actually calculate these types.
    10 years hard labour for crimes against mathematics, sport. \int sin^{-1}\;x \, dx etc. You don't believe me when I say it's important? ....... Consider:

    To find \int sin^{-1}\;x dx, use integration by parts: u = sin^{-1}\;x and dv = 1 dx
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by mr fantastic View Post
    10 years hard labour for crimes against mathematics, sport. \int sin^{-1}\;x \, dx etc. You don't believe me when I say it's important? ....... Consider:

    To find \int sin^{-1}\;x dx, use integration by parts: u = sin^{-1}\;x and dv = 1 dx
    Don't tell that to ThePerfectHacker. He hates the whole "dx" thing with a passion. Of course, anyone who has ever been able to understand that much of differential forms is probably a bit on the mad side. (I can say that legitimately because I am far further into the mad side of things than most others here! )

    -Dan
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    Quote Originally Posted by polymerase View Post
    \int \arcsin x\,dx
    You can also define f(x)=x\arcsin x. Contemplate its derivative and integrate.

    (It's actually integration by parts.)
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by topsquark View Post
    Don't tell that to ThePerfectHacker. He hates the whole "dx" thing with a passion. Of course, anyone who has ever been able to understand that much of differential forms is probably a bit on the mad side. (I can say that legitimately because I am far further into the mad side of things than most others here! )

    -Dan
    i dunno, i kinda like the whole dx thing.

    but yes, you are crazy
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Jhevon View Post
    i dunno, i kinda like the whole dx thing.

    but yes, you are crazy
    And I've got the prescriptions to prove it!

    -Dan
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    Quote Originally Posted by topsquark View Post
    Don't tell that to ThePerfectHacker. He hates the whole "dx" thing with a passion. Of course, anyone who has ever been able to understand that much of differential forms is probably a bit on the mad side. (I can say that legitimately because I am far further into the mad side of things than most others here! )

    -Dan
    Yes, I read that post (and the stuff between the lines) a while back. All good stuff.

    I was going to add for no fancy Dan (no pun intended ) to bring that higher level stuff to my notice - my opinion is it's horse for courses (and at the 1st yr undergraduate level the horses should be including the whole "dx" thing - further comment welcome). At the high school and 1st yr undergraduate level it's omission causes more trouble than it's inclusion, I think.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by mr fantastic View Post
    At the high school and 1st yr undergraduate level it's omission causes more trouble than it's inclusion, I think.
    I agree completely.

    -Dan
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    Quote Originally Posted by mr fantastic View Post
    Yes, I read that post (and the stuff between the lines) a while back. All good stuff.

    I was going to add for no fancy Dan (no pun intended ) to bring that higher level stuff to my notice - my opinion is it's horse for courses (and at the 1st yr undergraduate level the horses should be including the whole "dx" thing - further comment welcome). At the high school and 1st yr undergraduate level it's omission causes more trouble than it's inclusion, I think.
    Are you talking about this?
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    Quote Originally Posted by ThePerfectHacker View Post
    Are you talking about this?
    The same. A readable, accurate and comprehensive review.

    btw I'm sure it and your complex variables tutorials got you a vote for fancy latexer
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    Quote Originally Posted by mr fantastic View Post
    The same. A readable, accurate and comprehensive review.
    Thank you. I learned this from Functions of a Complex Variable I by John Conway, which I took out of the library. The book is nice, and advanced (GTM), but it has so many mistakes in it, I am scrippling in it to fix up its errors. That was the introductory chapter on complex integration.

    btw I'm sure it and your complex variables tutorials got you a vote for fancy latexer
    My LaTeX skills are basic. I do not even know why I got nominated. Oh, and I am returning to that tutorial. I did not have so much time*.


    *)I have said this before. One of my jobs is being an executioner. Currently, there are a lot of people who are sentenced to death. Preparation takes a long time. And thus I do not have so much time.
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