Need Help! Evaluate the Definite integral with U- substitution.

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Need Help! Evaluate the Definite integral with U- substitution.

Calculus1
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|dw:1439090251436:dw|
look up the derivative of \(\sin^{-1}x\)
substitute \(u=\sin^{-1}x\) then \(du=?\)

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Other answers:

I'm confused on how you got U to be sin-1x, how did you know it was that?
I thought U was supposed what's inside roots or paretheses, such as 1-x^2
do you know what the derivative of \(\sin^{-1}x\) is ?
Yea...it's 1/srt 1-x^2 * dx
that means substituting \(u = \sin^{-1}x\) simplifies the integrand, because \(du = \dfrac{1}{\sqrt{1-x^2}}dx\) that entire radical mess can be replaced by \(du\)
So I can't always assume that whats inside the parentheses or roots is always going to be substituted as U, right?
|dw:1439090695448:dw|
I get that!
No, with integrals there are no "strict" rules, you will have to do some guessing with each and every integral problem
okay thanks!
Also, don't forget to change the bounds accordingly
|dw:1439090839359:dw|
Why would I need to change the bounds?
because with u-substitution you're changing the variable of integration, so the bounds also change accordingly
|dw:1439090975245:dw|
those bounds refer to \(x\), they don't belong to \(u\)
|dw:1439091066367:dw|
|dw:1439091093678:dw|
no wait, whats the integrand right after substituting ?
|dw:1439091171917:dw|
Yes, whats antiderivative of \(u\) with respective to \(u\) ?
U(x)?
nope, whats antiderivative of \(x\) with respect to \(x\) ?
ohh....so would it be U^2/2
Yes
but isnt U supposed to be thought of a constant, therefore it's antiderivative should be U of x?
\(u\) is not constant, it is the variable that has replaced \(\sin^{-1}x\)
|dw:1439091490507:dw|
Looks good!
So do I get a decimal answer when I evaluate the integral?
plugin the bounds and see
Okay so I got (sin-1(1/2))^2/2? So do I box in that as the answer or its decimal...which is .137077
recall that \(\sin(\pi/6)=1/2\)
  • Ac3
look at the directions in the question
The directions in the questions just say evaluate the integral.
  • Ac3
if it asks for EXACT answers which most professors want then you give them that
  • Ac3
answer should be pi/6
  • Ac3
never do decimal unless it specifically tells you to. We know that arcsin of 1/2 is pi/6 because of the unit circle
ganeshie, is the answer (pi/6)^2/2?
  • Ac3
my bad pi/12
  • Ac3
wow i'm terrible today
hmm i got 5/2pi
haha thats not it
lol i need help then..
  • Ac3
the answer i gave is wrong it should be pi/72 i'll show simplification right now one sec.
nope thats still wrong
ah i got it pi^2/72
looks good! :)
Yes thats what I got too saseal!
  • Ac3
|dw:1439091910076:dw|
  • Ac3
i meant to say pi^2/72 originally
yes leave it as \(\dfrac{\pi^2}{72}\) do not convert to decimals unless asked to do so
Okay, cool, thanks ganeshie!
  • Ac3
Usually with u-substitution the rule of thumb is to try and get something to go away with your Du. That's where knowing you derivatives really well comes in to play. When in doubt just guess and check.
Thanks Ac3!

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