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anonymous
 5 years ago
Please, please help with substitution rule for indefinite integrals:
anonymous
 5 years ago
Please, please help with substitution rule for indefinite integrals:

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anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[\int\limits_{}^{} (4z + \sin(3z))e ^{6z ^{2}\cos(3z)} dz\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0whew, that's a doozy.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0So what do you think you want to use for substitution?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[\int\limits_{?}^{?} \sin(x) d x = \cos(x) + constant\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Not sure...possibly something to do with 3z? I have no idea to be honest

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I would let u = that yucky exponent on the e

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Okay but how does that help?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Well. \[u = 6x^2  cos(3z)\] \[\implies du = 12z +3sin(3z)dz = 3(4z + sin(3z))dz\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Oh so then \[du = 2z ^{3}\sin(3z)dx\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Close, but you integrated. You want to take the derivative

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Okay. Where do I plug u into?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Well first rewrite dz in terms of du using that last equation I had.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\(du = 3(4z + sin(3z))dz \implies dz =\ ?\)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I don't understand  what am I solving for?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Well for this part you're solving for dz.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Err...I divide 3(4z + sin(3z)) on both sides?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0So \[dz = \frac{1}{3(4z+sin(3z))} du\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0So now. Go back to your integral, and replace the exponent of e with u, and the dz with your expression for du, and you'll be pleasantly surprised.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Wait a minute...Why do I replace dz?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Because we are no longer integrating with respect to z, we are integrating with respect to u now. Because it turns out u is nicer.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0We are replacing our nasty expression with z for a nice expression with u.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[(4z + \sin(3z)e ^{u}\]du

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0But remember that dz does not equal du.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0ButI thought we replaced dz

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[dz = \frac{1}{3(4z+sin(3z))} du\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0So you have to put that whole thing in for dz

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Lets try an easy one to start with.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0So \[du = (2x)dx \implies dx = \frac{1}{2x}du\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Ok. So then \[\int x*e^{x^2}dx = \int x*e^u(\frac{1}{2x} du)\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0And we can cancel the x in front with the x in the denominator from the du.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0to get \[\frac{1}{2}\int e^udu\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Which is nice and easy.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[\frac{1}{2}\int e^udu = \frac{1}{2}e^u + c = \frac{1}{2}e^{x^2} + C\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0So I've got\[1/3 e ^{u}\] for my problem when do I integrate?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Before or after I plug in u

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Don't switch back your u until after you integrate.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0How do I integrate that?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[\int \frac{1}{3}e^u du = \frac{1}{3}\int e^udu\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0What is the integral of \(e^x\)?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[1/3 *e ^{6z ^{2}\cos(3z)} \]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Oh my goodness. Would you please walk me through one more?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Sure, but you'll have to do more of it this time ;)

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[\int\limits_{}^{}(4z ^{3} + 6\csc ^{2}(6z)(z ^{4}\cot(6z))^{4} dz\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Do I need to FOIL first?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0These are all ugly u substitution problems. Foiling will just make you cry.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0You want to find a nice thing to pick for a u substitution.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Do you notice anything interesting about these two things?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Not really sure what to look for, actually.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\((4z^3 + 6csc^2(6z)) * (z^4  cot(6z))^4\) You are looking for something in the equation that looks like the derivative of another part of the equation.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Err expression, not equation

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Oh! I guess the 4z^3 and z^4

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0And what about the \(6csc^2(6z)\) and the cot(6z) ?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0So what you want to do is pick the one that is not the derivative, and let that be u.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0So let\[u = z ^{4}  \cot(6z)\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Yep, then solve for dz in terms of z and du.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0So \[\int\limits_{}^{}(4z ^{3}+6\csc ^{2}(6z))u ^{4}du\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[u = z^4  cot(6z)\] So \[du =\ ?\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[du = (4z^3 + 6csc^2(6z))dz\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0So then what does dz = ?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0So plug in 1 over all that mess times du for dz in the original equation. and plug in u for what we said it equaled.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[(4z ^{3} + 6\csc ^{2}(6z))u ^{4} * 1/4z ^{3} + 6\csc ^{2}(6z)\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[(z ^{4}  \cot(6z))^{4}\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[\int(4z^3 + 6csc^2(6z)) * (z^4  cot(6z))^4dz\] \[ = \int (4z^3 + 6csc^2(6z)) u^4(\frac{1}{4z^3 + 6csc^2(6z)})du \] \[= \int u^4du =\ ?\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0I can't pull anything out in front of the integral sign

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[\int x^2dx = \frac{x^3}{3} +C\] Right?

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Yes. Once you have something nice, you just integrate.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0Then after you integrate you plug back in what u was.

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0\[(z ^{4}  \cot(6z)^{5}/ 5 +c\]

anonymous
 5 years ago
Best ResponseYou've already chosen the best response.0WOW. Good grief. Thanks so much. I am going to use these as examples for the next ones. Thanks again
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