Question

Hi. I trying to work out the integral of 1/(sqrt(2x-x^2)), but I'm moving too slowly. Can anyone give me any tips on how to move faster doing questions like this? @MIT 18.02 Multiva…

Answer 1

Write 2x - x^2 = 1 - (x-1)^2

Now substitute x - 1 = sin t

Answer 2

this looks like a setup for competing the square. since you are close with
\[2x-x^2\] i would change it to
... what jamesj said

Answer 3

but i would not make a trig sub
if you remember that
\[\frac{1}{\sqrt{1-x^2}}=\frac{d}{dx}\sin^{-1}(x)\] you are more than half way home

Answer 4

If you do that you should see that
\[\frac{1}{\sqrt{2x - x^2}} = \frac{1}{\sqrt{1 - \sin^2 t}} = \frac{1}{\cos t}\]

Answer 5

yet if you don't know that result this is how you prove it.

Answer 6

satellite likes to take the easy way out all the time

Answer 7

(or rather, this is one way to prove it.)

Answer 8

it never hurts to remember a few things. like the derivative of inverse trig functions (as least arcsine and arcosine) and derivative of roots and reciprocals. it is like knowing that 7 times 8 is 56 or whatever it is

Answer 9

i like making my students proving it each time
i figure they learn more that way and also their algebra sucks
so more algebra practice means they will get better at it right?

Answer 10

you are mean

Answer 11

formulas cheat them

Answer 12

you should make them prove each time that
\[\frac{d}{dx}x^n=nx^{n-1}\] because that is the only thing they are going to remember after 2 years trust me

Answer 13

lol
i tried to make them prove the product rule but thats hard for them

Answer 14

and then you should make them prove that
\[(f\circ g)'=f'(g)g'\] (not to bring up a sore subject...)

Answer 15

I actually this there's an important difference between the derivative of x^n and arcsin x. Which is how to deal with constants.

We all know what to do with the constants when integrating or differentiating cx^n

But unless you've done it recently, it's hard to remember exactly what the result is when you integrate
\[\frac{1}{\sqrt{a^2 - x^2}} \ \ or \frac{1}{\sqrt{1-a^2x^2}}\]

Answer 16

i will agree with that because unless i have done them recently i can integrate literally nothing