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anonymous
 one year ago
Show by hand how to find an antiderivative to
t E^t^2
anonymous
 one year ago
Show by hand how to find an antiderivative to t E^t^2

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jim_thompson5910
 one year ago
Best ResponseYou've already chosen the best response.0hint: usubsitution u = t^2, so du = 2t*dt

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.2\(\large\color{black}{\displaystyle\int\limits_{~}^{~}f'(x)\times e^{f(x)}~dx}\) SUBSTITUTION: `u=f(x) du=f'(x) dx` \(\large\color{black}{\displaystyle\int\limits_{~}^{~}e^{u}~du=e^u+C=e^{f(x)}+C}\) (this is an abstract case for some differentiable f(x) )

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0u =t^2 du = 2t dt E^u du = ...

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.2well, not exactly

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I suppose (1) can be pulled out

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.2\(\large\color{black}{\displaystyle\int\limits_{~}^{~}t\times e^{t^2}~dt}\) \(\large\color{black}{\displaystyle\int\limits_{~}^{~}t\times e^{t^2}~dt}\) \(\large\color{black}{\displaystyle u=t^2}\) \(\large\color{black}{\displaystyle du=2t~du~~~~\rightarrow~~~~\frac{1}{2}du=t~dt}\) \(\large\color{black}{\displaystyle\frac{1}{2}\int\limits_{~}^{~} e^{u}~du}\)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.2like that  same as you said, BUT you got the 1/2 there (why? I have wrote why in my post)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0and that 2 has to be removed I think.. du = 2t dt du/2 = 2t dt/2 du/2 = t dt ???

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.2okay, can you now tell me what your antiderivative would be?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.2(antiderivative of e^x is just e^x ... (well, +C) )

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.2One of my past professors took off most of the partial credit for +C. He explained that by saying that you gave only 1/∞ of all possible answers.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0thanks, nicely explained.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0my professors have been really good with that.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.2Enjoy it any time I'm on. Although sometimes I will be gone. I wish good luck to you in all, Yes, that's the way! lets make it roll 0~0

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0they had a grumble about how that happened to them.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.2If you have further questions, then u know  whenever I'm.... (or, there are many other math peeps here) Just always watch out how you solve for du in your substitution ....

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0actually if I had to use FTC with something like this equation.. for limits of [a,b] would they expect me to put +C on that too? say... 1/2 E^b^2  (1/2 E^a^2)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.2wait, what does FTC stand for?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0fundamental theorem calculus

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.2With limits of integration, +C ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I suppose it cancels out ...

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0unless there's some kind of initial condition

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.2If you are integrating a definite integral, you don't have the +C, because when you are integrating a definite integral you are finding the (numerical) area under the curve over a specific interval. And you are not finding the family of functions. So they say I guess: \(\large\color{black}{\displaystyle\int\limits_{a}^{b}f(x)~dx=\left( f(x)+C\right){\LARGE }^{x=b}_{x=a}}\) (where F'=f)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.2and yes C cancel's out, but it shouldn't be there in a first place.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.2+C is a family of antiderivative functions F(x) (and this family of F(x)+C is a set of functions that has a derivative of f) when finding area under the corve over some interval I , that is not a family of functions you are finding  that is an area calculation.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.2Not that the result would differ if you have +C there, since it will cancel, but as far as the concept goes +C, again, should not be there)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0gotcha.. makes sense
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