## anonymous one year ago Show by hand how to find an anti-derivative to -t E^t^2

1. SolomonZelman

u sub

2. SolomonZelman

u=t²

3. jim_thompson5910

hint: u-subsitution u = t^2, so du = 2t*dt

4. SolomonZelman

$$\large\color{black}{\displaystyle\int\limits_{~}^{~}f'(x)\times e^{f(x)}~dx}$$ 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) )

5. anonymous

u =t^2 du = 2t dt -E^u du = ...

6. SolomonZelman

well, not exactly

7. anonymous

I suppose (-1) can be pulled out

8. nincompoop

show by hand...

9. SolomonZelman

$$\large\color{black}{\displaystyle\int\limits_{~}^{~}-t\times e^{t^2}~dt}$$ $$\large\color{black}{\displaystyle-\int\limits_{~}^{~}t\times e^{t^2}~dt}$$ $$\large\color{black}{\displaystyle u=t^2}$$ $$\large\color{black}{\displaystyle du=2t~du~~~~\rightarrow~~~~\frac{1}{2}du=t~dt}$$ $$\large\color{black}{\displaystyle-\frac{1}{2}\int\limits_{~}^{~} e^{u}~du}$$

10. SolomonZelman

like that - same as you said, BUT you got the 1/2 there (why? I have wrote why in my post)

11. anonymous

and that 2 has to be removed I think.. du = 2t dt du/2 = 2t dt/2 du/2 = t dt ???

12. anonymous

thanks solomon...

13. SolomonZelman

okay, can you now tell me what your antiderivative would be?

14. SolomonZelman

(antiderivative of e^x is just e^x ... (well, +C) )

15. anonymous

-1/2 E^t^2

16. SolomonZelman

yes, with +C

17. anonymous

oh yes.. +C

18. SolomonZelman

One 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.

19. anonymous

thanks, nicely explained.

20. anonymous

my professors have been really good with that.

21. SolomonZelman

Enjoy 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

22. anonymous

23. SolomonZelman

If 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 ....

24. anonymous

actually 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)

25. SolomonZelman

wait, what does FTC stand for?

26. anonymous

fundamental theorem calculus

27. SolomonZelman

oh.

28. SolomonZelman

With limits of integration, +C ?

29. anonymous

I suppose it cancels out ...

30. anonymous

unless there's some kind of initial condition

31. SolomonZelman

If 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)

32. SolomonZelman

and yes C cancel's out, but it shouldn't be there in a first place.

33. SolomonZelman

+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.

34. SolomonZelman

Not 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)

35. anonymous

gotcha.. makes sense

36. SolomonZelman

Alrighty:)

37. anonymous

laterzs

38. anonymous

thanks again

39. SolomonZelman

Anytime