anonymous
  • anonymous
Taking the derivative of an integral...
Mathematics
  • Stacey Warren - Expert brainly.com
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chestercat
  • chestercat
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anonymous
  • anonymous
How do I take the derivative of the function \[g(x)=\int\limits_{x}^{x^2}(t^2-5)/(t^2+2)dt\]
amistre64
  • amistre64
the derivative and an integral are inverse operations: thed cancel each other out and youd end up with: g'(x) = (t^2-5)/(t^2+2) ..... as far as I recollect
anonymous
  • anonymous
How exactly does it cancel out?

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amistre64
  • amistre64
hmmm..... you might have a point, I failed to recognize the variables....
anonymous
  • anonymous
So.. how would you do it?
amistre64
  • amistre64
take the integral, and substitute for t with x...... then derive it back down..... but the actuall process eludes me :)
amistre64
  • amistre64
can we get the integrand into a sum of terms?
amistre64
  • amistre64
dunno if a trig substitution would be useful: t = sqrt(2) tan(a)
anonymous
  • anonymous
Idk.. I'm just totally lost on this question. So I substitute the t's for x's?
anonymous
  • anonymous
Or do I substitute the t^2 in?
amistre64
  • amistre64
my limited knowledge on this would suggest that I find g(x) from the integrand first.... an exact HOW, is beyind me at the moment :)
anonymous
  • anonymous
Lol ok. Thank you for your help though :)
amistre64
  • amistre64
if I did it right in my thoguths here: the new integrand becomes: tan^2(a) -5/2 which looks easier to work with...
anonymous
  • anonymous
nicole did you finish most of the packet? i didn't...i do have a much geater understanding about integrals though :)
anonymous
  • anonymous
No.. I think I'm gonna stay after school.
anonymous
  • anonymous
mother just told me she is picking me up. i have NO idea why. i want to stay too..
amistre64
  • amistre64
i substituted: t = sqrt(2) tan(a) ; dt = sqrt(2) sec^2 2tan^2(a) -5 sqrt(2)sec^2(a) da ------------- ------------- 2(tan^2(a) + 1) 1 2tan^2(a) -5 sqrt(2)sec^2(a) da ------------- ------------- 2sec^2(a) 1 2tan^2(a) -5 sqrt(2) da ------------- ------- 2 1 2tan^2(a) -5 sqrt(2) da ---------- - --- ------- 2 2 1 sqrt(2) tan^2(a) - 5sqrt(2)/2 da is what I get actually, but it might be easier to integrate
anonymous
  • anonymous
Just stay.. find a ride home.
anonymous
  • anonymous
Wow.. That's a whole lot..
amistre64
  • amistre64
or since the sqrt(2) is a constant, we can yank it out to the other side: sqrt(2) [S] tan^2(a) - 5/2 da
amistre64
  • amistre64
myins here....now ill now what Im doing wrong ;)
myininaya
  • myininaya
Let f(t)=(t^2-5)/(t^2+2) Then int(f(t), t=x..x^2)=F(x^2)-F(x) To take derivative of this. We have to use chain rule. Therefore g'(x)=2xf(x^2)-f(x)
myininaya
  • myininaya
g'(x)=2x[(x^4-5)/(x^4+2)]-(x^2-5)/(x^2+2)
myininaya
  • myininaya
any questions?
myininaya
  • myininaya
we dont need to integrate to find the derivative of the integral
amistre64
  • amistre64
......thats what I thought :) but those x..x^2 things threw me for a loop
myininaya
  • myininaya
lol where did you get all of those fans? I'm so jealous
amistre64
  • amistre64
they were having a sale at pier1 ;)
myininaya
  • myininaya
damn i missed it :(
myininaya
  • myininaya
i think he lefted :(
amistre64
  • amistre64
maybe, but the math remained lol
myininaya
  • myininaya
true im going to go play call of duty now peace

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