anonymous
  • anonymous
how do we integrate z/(x^2+z^2) with respect to x the answer is arctan(x/z)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
\[\int\limits_{}^{}\frac{ z }{ x ^{2}+z ^{2} }dx\]
anonymous
  • anonymous
this is a formula you use to compute the integral of a function with this structure.
anonymous
  • anonymous
You will be able to relate your question simply to the standard integral \[\large \frac{1 }{ a^2+x^2 } =\frac{1 }{a } \times \arctan (\frac{x}{ a })\]

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anonymous
  • anonymous
if Left hand side is the function that has to integrate with respect to x
anonymous
  • anonymous
is 1/a the derivative of x/a
anonymous
  • anonymous
is tht why its there?
anonymous
  • anonymous
1/a is part of the standard formula. I'll work it out for you.
anonymous
  • anonymous
\[z \int\limits_{}^{}\frac{ 1 }{ x ^{2}+z ^{2} }dx \]
anonymous
  • anonymous
thnk you id like to see why
anonymous
  • anonymous
\[z*\frac{ 1 }{ z }\arctan(\frac{ x }{ z })\rightarrow \arctan(\frac{ x }{ z })\]
anonymous
  • anonymous
in this case z is a scalar.
anonymous
  • anonymous
so typically z would be represented as a number. I think they are using zahlen which denotes integer values.
anonymous
  • anonymous
why is arc tan (x/z) and where does the 1/z come from??
anonymous
  • anonymous
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.
anonymous
  • anonymous
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 ??
anonymous
  • anonymous
it is 1/1+x^2
anonymous
  • anonymous
well,good.. then what will be derivative of arctan(x/a)
anonymous
  • anonymous
i dunno
anonymous
  • anonymous
ok.. instead of x it is x/a.. substitute that value in your answer above
anonymous
  • anonymous
and rember x/a is another function so you need to apply the chain rule of differentiation
anonymous
  • anonymous
1/(1+(x/a)^2)*(1/a)
anonymous
  • anonymous
Yup........... You got it.. simplify it
anonymous
  • anonymous
I mean, I can do the integral if you want it in all the gruesome detail...
anonymous
  • anonymous
a/(x2+a^2)
anonymous
  • anonymous
yea shure if its not too complicated
anonymous
  • anonymous
well you are right ... now compare with your question......you got your answer
anonymous
  • anonymous
that is you got \[\frac{ d }{ dx }(\arctan(x/a) = \frac{ a }{ x^2+a^2 }\]
anonymous
  • anonymous
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) :/
anonymous
  • anonymous
now itegrat on both sides.. you got answer
anonymous
  • anonymous
Remember Integration is the opposite of differentiation @SABREEN .. now what will be tha answer?
anonymous
  • anonymous
so than the integral of a/(x^2+a^2) is arctan (x/a)
anonymous
  • anonymous
thanks=)
anonymous
  • anonymous
yes.. Welcome.. Now should I explain how 1/a comes ???? ;)
anonymous
  • anonymous
Next one in line: \[\int\limits \tanh^{-1}(ax)dx\]
anonymous
  • anonymous
loll nahh i got this bro=)
anonymous
  • anonymous
@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.. :)
anonymous
  • anonymous
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)

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