Simple calculus question

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Simple calculus question

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

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