Question

Integration by parts problem: sin^-1(x)

Answer 1

I'm just not really sure how to get started with this problem: \[\int\limits_{?}^{?}\sin^{-1} x dx\]It's supposed to be integration by parts, so I rewrote it as \[\int\limits_{?}^{?}\frac{1}{\sin x}dx\]

Answer 2

but not sure where to go from there...

Answer 3

Nooooooooo!\[\Large\bf\sf \sin^{-1}x\quad\ne\quad \frac{1}{\sin x}\]

Answer 4

We never use the -1 for powers of trig functions.
That is reserved for the inverse function.\[\Large\bf\sf \sin^{-1}x\quad=\quad \arcsin x\]

Answer 5

That's what I thought, but I don't see how you could possibly integrate arcsin(x) by parts...it seems like it would just be like sin(x) where you memorize it...

Answer 6

\[\sin^{-1}(x)=u, dx = dv, du =\frac{1}{\sqrt{1-x^2}}, v=x\]

Answer 7

Well, if you could make a shortcut, taking the derivative of arcsin, it would turn it into a bunch of stuff involving x right?

That's what by-parts will do for us.

Answer 8

Ohh...dang I am feeling dumb now...thanks.

Answer 9

if i were you, once i got the answer i would memorize it
that way if you see it on a test or a quiz, while your colleagues are doing all the work, you can just write down the answer
same trick as for
\[\int \log(x)dx\] parts gives \[x\log(x)-x\] another good one to memorize