anonymous one year ago Simple calculus question

1. nincompoop

So simple it has no answer

2. anonymous

Im stuck on a specific type of question, can someone help me?

3. anonymous

4. amistre64

do you know the integration of sec(t) ?

5. anonymous

Why is that necessary?

6. amistre64

its not really, but its a good check ...

7. anonymous

Im not quite sure where sec come into this.

8. amistre64

sec and csc have similar derivatives i recall sec and then adjust for csc

9. amistre64

the integration of sec was well known prior to the founding of integration :)

10. anonymous

The derivative of csc i believe is -cscxcotx

11. amistre64

the derivative yes, but the integration? its a sneaky little trick

12. amistre64

$\frac{csc}{1}*\frac{-csc-cot}{-csc-cot}$ $\frac{-csc^2-csc~cot}{-csc-cot}\implies~ln(csc+cot)$

13. anonymous

Im lost, simply how do you go from a derivative to an integral which is equal to the original function. Is it as simple as moving the derivative into the integral?

14. anonymous

It's the fundamental theorem of calculus $\frac{ d }{ dx } \int\limits_{a}^{x} f(t) dt = f(x)$

15. amistre64

we have 2 options, we can use the fundamental thrm: $\int_{a}^{x}f'(t)dt=F(x)-F(a)$ and take the derivative $\frac d{dx}\int_{a}^{x}f'(t)dt=f'(x)x'-f'(a)a'$

16. anonymous

So the derivative of the integral of a function in that format is equal to that function.. so since i have the derivative do i find the anti-derivative and plug that into the integral?

17. amistre64

or we can work the integration, and then take the derivative, either way

18. anonymous

My course has recently worked with the fundamental therom so I assume thats the way they want me to solve it.

19. amistre64

you asked about which one it shouldbe, im just suggesting that in a pinch, you can work the long way if possible

20. anonymous

So... does that make the answer b or am i on the wrong route?

21. anonymous

It seems to me this problem is designed to be simple im just making it difficult.

22. amistre64

it is designed to make you apply the fundamental thrm yes

23. anonymous

Hey sorry, how did you get $\frac{-\csc^2-\csc~\cot}{-\csc-\cot}\implies~\ln(\csc+\cot)$

24. anonymous

Hmm?

25. amistre64

prolly an error in my head, thinkig to quick -ln(csc + cot) derives to csc like i said, im used to the sec form :)

26. amistre64

x should be the high range, so its either the first or last option to me

27. anonymous

Or b

28. anonymous

the negation flips the limits, correct?

29. anonymous

never mind

30. anonymous

I know its not c (duh). I dont believe it is a. However i dont know whether i put the antiderivtive of the derivitive into the intergral or the derivative into the integral

31. anonymous

IE, b or d

32. amistre64

-ln(csc(x)+cot(x)) + C derives to csc(x) since y = -ln(csc(x)+cot(x)) + C, i dont see why we would have a constant in the integration when dy/dx = csc(x)

33. amistre64

a or d is my thought

34. anonymous

the constant i believe has something to do with making the y value correct, they all have them so i think its correct.

35. anonymous

A or D was what i started stuck between XD

36. anonymous

im sorry b or d

37. amistre64

$y=\int y'(t)~dt +C$ i see it now, these old eyes were placing it inside the dt

38. anonymous

XD thats a correct therom? so that would make it d?

39. anonymous

Essentially there youre using the intergral to find the antiderivative?

40. amistre64

lets do the long way and check it out $y=\int_{a}^{x}csc(t)~dt+C=-ln(csc(x)+cot(x))+ln(csc(a)+cot(a))+C~$ when x=4, y=-9 $-9+ln(csc(4)+cot(4))=ln(csc(a)+cot(a))+C$

41. amistre64

im using the integral to determine that solution yes :)

42. amistre64

let a=4, and C=-9

43. anonymous

Thanks :) It really was simple XD

44. anonymous

I figured

45. amistre64

notice that by comparing like parts: ln(csc(4)+cot(4)) = ln(csc(a)+cot(a)) , let a=4 -9 = C ...

46. amistre64

another way to have viewed this, now that i have my bearings straight might be like this: 4 is in the domain element, it gets used as an input value ... it should be in the limit interval -9 is a range element, its not necessarily a part of the integrals domain limits

47. amistre64

but thats just a guess at what i see, i would not have determined that by just the FTC

48. anonymous

Thanks: i have actually solved this equation before (and whtat you are saying sounds farmilar) but i was dead tired and couldnnt remember how i did it.

49. amistre64

good luck :)

50. anonymous

not to butt in, but why isn't the answer obviously 4?

51. anonymous

Haha I was thinking the same thing, but anyway it was a nice question thanks @sccitestla and thanks @amistre64 :)