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Evaluate the Definite integral with U- substitution.

- anonymous

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Evaluate the Definite integral with U- substitution.

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- anonymous

|dw:1439090251436:dw|

- ganeshie8

look up the derivative of \(\sin^{-1}x\)

- ganeshie8

substitute \(u=\sin^{-1}x\)
then \(du=?\)

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## More answers

- anonymous

I'm confused on how you got U to be sin-1x, how did you know it was that?

- anonymous

I thought U was supposed what's inside roots or paretheses, such as 1-x^2

- ganeshie8

do you know what the derivative of \(\sin^{-1}x\) is ?

- anonymous

Yea...it's 1/srt 1-x^2 * dx

- ganeshie8

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 \(du\)

- anonymous

So I can't always assume that whats inside the parentheses or roots is always going to be substituted as U, right?

- ganeshie8

|dw:1439090695448:dw|

- anonymous

I get that!

- ganeshie8

No, with integrals there are no "strict" rules, you will have to do some guessing with each and every integral problem

- anonymous

okay thanks!

- ganeshie8

Also, don't forget to change the bounds accordingly

- ganeshie8

|dw:1439090839359:dw|

- anonymous

Why would I need to change the bounds?

- ganeshie8

because with u-substitution you're changing the variable of integration, so the bounds also change accordingly

- ganeshie8

|dw:1439090975245:dw|

- ganeshie8

those bounds refer to \(x\),
they don't belong to \(u\)

- anonymous

|dw:1439091066367:dw|

- anonymous

|dw:1439091093678:dw|

- ganeshie8

no wait, whats the integrand right after substituting ?

- anonymous

|dw:1439091171917:dw|

- ganeshie8

Yes, whats antiderivative of \(u\) with respective to \(u\) ?

- anonymous

U(x)?

- ganeshie8

nope, whats antiderivative of \(x\) with respect to \(x\) ?

- anonymous

ohh....so would it be U^2/2

- ganeshie8

Yes

- anonymous

but isnt U supposed to be thought of a constant, therefore it's antiderivative should be U of x?

- ganeshie8

\(u\) is not constant, it is the variable that has replaced \(\sin^{-1}x\)

- anonymous

|dw:1439091490507:dw|

- ganeshie8

Looks good!

- anonymous

So do I get a decimal answer when I evaluate the integral?

- ganeshie8

plugin the bounds and see

- anonymous

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

- ganeshie8

recall that \(\sin(\pi/6)=1/2\)

- Ac3

look at the directions in the question

- anonymous

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

- anonymous

ganeshie, is the answer (pi/6)^2/2?

- Ac3

my bad pi/12

- Ac3

wow i'm terrible today

- anonymous

hmm i got 5/2pi

- ganeshie8

haha thats not it

- anonymous

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.

- ganeshie8

nope thats still wrong

- anonymous

ah i got it pi^2/72

- ganeshie8

looks good! :)

- anonymous

Yes thats what I got too saseal!

- Ac3

|dw:1439091910076:dw|

- Ac3

i meant to say pi^2/72 originally

- ganeshie8

yes leave it as \(\dfrac{\pi^2}{72}\)
do not convert to decimals unless asked to do so

- anonymous

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.

- anonymous

Thanks Ac3!

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