Thread: Intergrals of Inverse Trig Functions

1. 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: $\displaystyle \int sin^{-1}\;x$ or $\displaystyle \int cos^{-1}\;x$ etc. Can anyone show me how to actually calculate these types.

2. Originally Posted by polymerase
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: $\displaystyle \int sin^{-1}\;x$ or $\displaystyle \int cos^{-1}\;x$ 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

3. Originally Posted by polymerase
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: $\displaystyle \int sin^{-1}\;x$ or $\displaystyle \int cos^{-1}\;x$ etc. Can anyone show me how to actually calculate these types.
Do it by parts. For example
$\displaystyle \int sin^{-1}(x)~dx = x~sin^{-1}(x) - \int \frac{x}{\sqrt{1 - x^2}}~dx$

-Dan

Edit: Is there an echo in here?

4. Originally Posted by polymerase
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: $\displaystyle \int sin^{-1}\;x$ or $\displaystyle \int cos^{-1}\;x$ etc. Can anyone show me how to actually calculate these types.
10 years hard labour for crimes against mathematics, sport. $\displaystyle \int sin^{-1}\;x \,$ dx etc. You don't believe me when I say it's important? ....... Consider:

To find $\displaystyle \int sin^{-1}\;x$ dx, use integration by parts: $\displaystyle u = sin^{-1}\;x$ and $\displaystyle dv = 1$ dx

5. Originally Posted by mr fantastic
10 years hard labour for crimes against mathematics, sport. $\displaystyle \int sin^{-1}\;x \,$ dx etc. You don't believe me when I say it's important? ....... Consider:

To find $\displaystyle \int sin^{-1}\;x$ dx, use integration by parts: $\displaystyle u = sin^{-1}\;x$ and $\displaystyle dv = 1$ 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

6. Originally Posted by polymerase
$\displaystyle \int \arcsin x\,dx$
You can also define $\displaystyle f(x)=x\arcsin x.$ Contemplate its derivative and integrate.

(It's actually integration by parts.)

7. Originally Posted by topsquark
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

8. Originally Posted by Jhevon
i dunno, i kinda like the whole dx thing.

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

-Dan

9. Originally Posted by topsquark
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.

10. Originally Posted by mr fantastic
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

11. Originally Posted by mr fantastic
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.

12. Originally Posted by ThePerfectHacker
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

13. Originally Posted by mr fantastic
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.