anonymous
  • anonymous
Simple calculus question
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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nincompoop
  • nincompoop
So simple it has no answer
anonymous
  • anonymous
Im stuck on a specific type of question, can someone help me?
anonymous
  • anonymous
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amistre64
  • amistre64
do you know the integration of sec(t) ?
anonymous
  • anonymous
Why is that necessary?
amistre64
  • amistre64
its not really, but its a good check ...
anonymous
  • anonymous
Im not quite sure where sec come into this.
amistre64
  • amistre64
sec and csc have similar derivatives i recall sec and then adjust for csc
amistre64
  • amistre64
the integration of sec was well known prior to the founding of integration :)
anonymous
  • anonymous
The derivative of csc i believe is -cscxcotx
amistre64
  • amistre64
the derivative yes, but the integration? its a sneaky little trick
amistre64
  • amistre64
\[\frac{csc}{1}*\frac{-csc-cot}{-csc-cot}\] \[\frac{-csc^2-csc~cot}{-csc-cot}\implies~ln(csc+cot)\]
anonymous
  • 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?
anonymous
  • anonymous
It's the fundamental theorem of calculus \[\frac{ d }{ dx } \int\limits_{a}^{x} f(t) dt = f(x)\]
amistre64
  • 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 \[\frac d{dx}\int_{a}^{x}f'(t)dt=f'(x)x'-f'(a)a'\]
anonymous
  • 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?
amistre64
  • amistre64
or we can work the integration, and then take the derivative, either way
anonymous
  • anonymous
My course has recently worked with the fundamental therom so I assume thats the way they want me to solve it.
amistre64
  • amistre64
you asked about which one it shouldbe, im just suggesting that in a pinch, you can work the long way if possible
anonymous
  • anonymous
So... does that make the answer b or am i on the wrong route?
anonymous
  • anonymous
It seems to me this problem is designed to be simple im just making it difficult.
amistre64
  • amistre64
it is designed to make you apply the fundamental thrm yes
anonymous
  • anonymous
Hey sorry, how did you get \[\frac{-\csc^2-\csc~\cot}{-\csc-\cot}\implies~\ln(\csc+\cot)\]
anonymous
  • anonymous
Hmm?
amistre64
  • 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 :)
amistre64
  • amistre64
x should be the high range, so its either the first or last option to me
anonymous
  • anonymous
Or b
anonymous
  • anonymous
the negation flips the limits, correct?
anonymous
  • anonymous
never mind
anonymous
  • 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
anonymous
  • anonymous
IE, b or d
amistre64
  • 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)
amistre64
  • amistre64
a or d is my thought
anonymous
  • anonymous
the constant i believe has something to do with making the y value correct, they all have them so i think its correct.
anonymous
  • anonymous
A or D was what i started stuck between XD
anonymous
  • anonymous
im sorry b or d
amistre64
  • amistre64
\[y=\int y'(t)~dt +C\] i see it now, these old eyes were placing it inside the dt
anonymous
  • anonymous
XD thats a correct therom? so that would make it d?
anonymous
  • anonymous
Essentially there youre using the intergral to find the antiderivative?
amistre64
  • 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 \[-9+ln(csc(4)+cot(4))=ln(csc(a)+cot(a))+C\]
amistre64
  • amistre64
im using the integral to determine that solution yes :)
amistre64
  • amistre64
let a=4, and C=-9
anonymous
  • anonymous
Thanks :) It really was simple XD
anonymous
  • anonymous
I figured
amistre64
  • amistre64
notice that by comparing like parts: ln(csc(4)+cot(4)) = ln(csc(a)+cot(a)) , let a=4 -9 = C ...
amistre64
  • 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
amistre64
  • amistre64
but thats just a guess at what i see, i would not have determined that by just the FTC
anonymous
  • 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.
amistre64
  • amistre64
good luck :)
anonymous
  • anonymous
not to butt in, but why isn't the answer obviously 4?
anonymous
  • anonymous
Haha I was thinking the same thing, but anyway it was a nice question thanks @sccitestla and thanks @amistre64 :)

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