#Integration
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elder robin
$$ \frac{1}{t^2}\int _0^\infty \frac{1}{1+ \left(\frac{x}{t}\right)^2}dx $$
inland rampartBOT
aL
elder robin
this should look familiar
still bluff
ur faster than me
shadow trellis
1/1+x^2
elder robin
if the denominator is in a given power then as it says, use the Leibniz rule
don't try to find explicit antiderivatives
verify that you can differentiate under integral and differentiate
$$ \int _0^\infty \frac{1}{2(x^2+t^2)2t}dx $$
inland rampartBOT
aL
shadow trellis
isee hat u taken out the t^2 but where did the power 2 go
inland rampartBOT
aL
elder robin
then if some nice conditions are satisfied we have
$$ I'(t) = \int _0^\infty \frac{\partial}{\partial t} \frac{1}{(x^2+t^2)^2}dx $$
inland rampartBOT
aL
elder robin
$$ \frac{1}{t}\left( \frac{\pi}{2} - 0 \right) $$
inland rampartBOT
aL
elder robin
$$ \lim _{x\to\infty} \arctan x = \frac{\pi}{2} $$
inland rampartBOT
aL
shadow trellis
but you said you did i)
i got it. do we need to assign d/dx to the integration ourselves? because the question didn't provide
shadow trellis
let's say, if the question is integrate 3x dx, then by using leibniz rule, can we rewrite the expression as d/dx integrate 3x dx (we add the d/dx ourselves)
elder robin
to show you that this is wrong
elder robin
let's say, if the question is integrate 3x dx, then by using leibniz rule, can w...
d/dx? if you have to integrate 3x then it's integral 3x dx, that's it
shadow trellis
where did i did mistake
elder robin
re-evaluate this limit
shadow trellis
d/dx? if you have to integrate 3x then it's integral 3x dx, that's it
this is just an example, the 3x in this case is the expression in ii)
elder robin
$$ = \frac{1}{t} \lim _{a\to\infty} \arctan \frac{a}{t} - \arctan 0 = \frac{\pi}{2t}$$
inland rampartBOT
aL
elder robin
this is just an example, the 3x in this case is the expression in ii)
it is not a parametrised integral
very different situation
elder robin
let's say, if the question is integrate 3x dx, then by using leibniz rule, can w...
are you talking about this?
shadow trellis
ya
elder robin
that's not d/dx
elder robin
this is the first step for i) ?
that's the answer of i)
you wrote something different as the answer
this is a nonnegative integral
but you write that the result is this
which is negative
this is impossible
shadow trellis
ik that it is an asymptote but idk how to show the step out
elder robin
are you paying attention
shadow trellis
so sorry i am still learning the basics of calculus, i have problem understanding difficult terms that you have mentioned above, please forgive my lack of knowledge and i will try my very best to get what you are saying, thank you for your patience
elder robin
$$I'(t) = \int _0^\infty \frac{\partial}{\partial t} \frac{1}{(x^2+t^2)^2}dx = \int _0^\infty \frac{1}{2(x^2+t^2)2t}dx $$
inland rampartBOT
aL
elder robin
$$ = \frac{1}{4t} \int _0^\infty \frac{1}{x^2+t^2}dx $$
inland rampartBOT
aL
elder robin
what do you think this is?
shadow trellis
first step
elder robin
what?
shadow trellis
then i substitute ans for i into it
shadow trellis
pi/8t^2
elder robin
correct
shadow trellis
ok tqvm appreciate it
elder robin
now this
shadow trellis
i differentiate it then sub the pi/8t^2 in it
elder robin
yes!
$$ \int _0^\infty \frac{\partial}{\partial t} \frac{1}{(x^2+t^2)^3}dx = \int _0^\infty \frac{1}{3(x^2+t^2)^22t}dx = \frac{1}{6t} \int _0^\infty \frac{1}{(x^2+t^2)^2}dx $$
inland rampartBOT
aL
shadow trellis
ok tqvm
elder robin
are you familiar with induction?
shadow trellis
what is induction
shadow trellis
i understand the word but idk what is it for
elder robin
ok this is above my paygrade then
you need to apply mathematical induction to do iii)
shadow trellis
ok i will try it out myself tq
elder robin
shadow trellis
ember cairn
ok i will try it out myself tq
https://youtu.be/x5cWX-EyLEI?si=NHGSBNHfyq_3yIyH this guy explains it pretty well, hope you find it helpful
knotty gustBOT
@shadow trellis
half jolt
+close