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how do we integrate z/(x^2+z^2) with respect to x
the answer is arctan(x/z)
 one year ago
 one year ago
how do we integrate z/(x^2+z^2) with respect to x the answer is arctan(x/z)
 one year ago
 one year ago

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VeritasVosLiberabitBest ResponseYou've already chosen the best response.0
\[\int\limits_{}^{}\frac{ z }{ x ^{2}+z ^{2} }dx\]
 one year ago

VeritasVosLiberabitBest ResponseYou've already chosen the best response.0
this is a formula you use to compute the integral of a function with this structure.
 one year ago

Rosh007Best ResponseYou've already chosen the best response.1
You will be able to relate your question simply to the standard integral \[\large \frac{1 }{ a^2+x^2 } =\frac{1 }{a } \times \arctan (\frac{x}{ a })\]
 one year ago

Rosh007Best ResponseYou've already chosen the best response.1
if Left hand side is the function that has to integrate with respect to x
 one year ago

SABREENBest ResponseYou've already chosen the best response.1
is 1/a the derivative of x/a
 one year ago

VeritasVosLiberabitBest ResponseYou've already chosen the best response.0
1/a is part of the standard formula. I'll work it out for you.
 one year ago

VeritasVosLiberabitBest ResponseYou've already chosen the best response.0
\[z \int\limits_{}^{}\frac{ 1 }{ x ^{2}+z ^{2} }dx \]
 one year ago

SABREENBest ResponseYou've already chosen the best response.1
thnk you id like to see why
 one year ago

VeritasVosLiberabitBest ResponseYou've already chosen the best response.0
\[z*\frac{ 1 }{ z }\arctan(\frac{ x }{ z })\rightarrow \arctan(\frac{ x }{ z })\]
 one year ago

VeritasVosLiberabitBest ResponseYou've already chosen the best response.0
in this case z is a scalar.
 one year ago

VeritasVosLiberabitBest ResponseYou've already chosen the best response.0
so typically z would be represented as a number. I think they are using zahlen which denotes integer values.
 one year ago

SABREENBest ResponseYou've already chosen the best response.1
why is arc tan (x/z) and where does the 1/z come from??
 one year ago

VeritasVosLiberabitBest ResponseYou've already chosen the best response.0
1/z is always part of the equation. if you want me to do a proof, I'm afraid I don't have enough time for that but maybe someone else would.
 one year ago

Rosh007Best ResponseYou've already chosen the best response.1
the formula for most of the integral is coming fromits derivative remember derivation is the opposite of integration and wise versa. can you say what is th derivative of f(x) = arctan x ??
 one year ago

Rosh007Best ResponseYou've already chosen the best response.1
well,good.. then what will be derivative of arctan(x/a)
 one year ago

Rosh007Best ResponseYou've already chosen the best response.1
ok.. instead of x it is x/a.. substitute that value in your answer above
 one year ago

Rosh007Best ResponseYou've already chosen the best response.1
and rember x/a is another function so you need to apply the chain rule of differentiation
 one year ago

Rosh007Best ResponseYou've already chosen the best response.1
Yup........... You got it.. simplify it
 one year ago

malevolence19Best ResponseYou've already chosen the best response.0
I mean, I can do the integral if you want it in all the gruesome detail...
 one year ago

SABREENBest ResponseYou've already chosen the best response.1
yea shure if its not too complicated
 one year ago

Rosh007Best ResponseYou've already chosen the best response.1
well you are right ... now compare with your question......you got your answer
 one year ago

Rosh007Best ResponseYou've already chosen the best response.1
that is you got \[\frac{ d }{ dx }(\arctan(x/a) = \frac{ a }{ x^2+a^2 }\]
 one year ago

malevolence19Best ResponseYou've already chosen the best response.0
I mean, there isn't much too it though. The formula is basically where it comes from... Now that I think about it, I don't know how much you can actually write. I was running it by wolfram and all they do is a variable substitution and then use that int (1/u^2+1)=arctan(u) :/
 one year ago

Rosh007Best ResponseYou've already chosen the best response.1
now itegrat on both sides.. you got answer
 one year ago

Rosh007Best ResponseYou've already chosen the best response.1
Remember Integration is the opposite of differentiation @SABREEN .. now what will be tha answer?
 one year ago

SABREENBest ResponseYou've already chosen the best response.1
so than the integral of a/(x^2+a^2) is arctan (x/a)
 one year ago

Rosh007Best ResponseYou've already chosen the best response.1
yes.. Welcome.. Now should I explain how 1/a comes ???? ;)
 one year ago

malevolence19Best ResponseYou've already chosen the best response.0
Next one in line: \[\int\limits \tanh^{1}(ax)dx\]
 one year ago

SABREENBest ResponseYou've already chosen the best response.1
loll nahh i got this bro=)
 one year ago

Rosh007Best ResponseYou've already chosen the best response.1
@SABREEN ....Good... Welcome and enjoy.... @malevolence19 .......you could solve by following the method i described here...only difference is x = ax and remember ax is a fuction.. :)
 one year ago

malevolence19Best ResponseYou've already chosen the best response.0
I mean, I know how to to do it lol. It's a integration by parts problem once you look up the derivative to tanh^(1)(x)
 one year ago
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