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

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

VeritasVosLiberabit
 2 years ago
Best ResponseYou've already chosen the best response.0this is a formula you use to compute the integral of a function with this structure.

Rosh007
 2 years ago
Best ResponseYou've already chosen the best response.1You will be able to relate your question simply to the standard integral \[\large \frac{1 }{ a^2+x^2 } =\frac{1 }{a } \times \arctan (\frac{x}{ a })\]

Rosh007
 2 years ago
Best ResponseYou've already chosen the best response.1if Left hand side is the function that has to integrate with respect to x

SABREEN
 2 years ago
Best ResponseYou've already chosen the best response.1is 1/a the derivative of x/a

VeritasVosLiberabit
 2 years ago
Best ResponseYou've already chosen the best response.01/a is part of the standard formula. I'll work it out for you.

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

SABREEN
 2 years ago
Best ResponseYou've already chosen the best response.1thnk you id like to see why

VeritasVosLiberabit
 2 years ago
Best ResponseYou've already chosen the best response.0\[z*\frac{ 1 }{ z }\arctan(\frac{ x }{ z })\rightarrow \arctan(\frac{ x }{ z })\]

VeritasVosLiberabit
 2 years ago
Best ResponseYou've already chosen the best response.0in this case z is a scalar.

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

SABREEN
 2 years ago
Best ResponseYou've already chosen the best response.1why is arc tan (x/z) and where does the 1/z come from??

VeritasVosLiberabit
 2 years ago
Best ResponseYou've already chosen the best response.01/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.

Rosh007
 2 years ago
Best ResponseYou've already chosen the best response.1the 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 ??

Rosh007
 2 years ago
Best ResponseYou've already chosen the best response.1well,good.. then what will be derivative of arctan(x/a)

Rosh007
 2 years ago
Best ResponseYou've already chosen the best response.1ok.. instead of x it is x/a.. substitute that value in your answer above

Rosh007
 2 years ago
Best ResponseYou've already chosen the best response.1and rember x/a is another function so you need to apply the chain rule of differentiation

Rosh007
 2 years ago
Best ResponseYou've already chosen the best response.1Yup........... You got it.. simplify it

malevolence19
 2 years ago
Best ResponseYou've already chosen the best response.0I mean, I can do the integral if you want it in all the gruesome detail...

SABREEN
 2 years ago
Best ResponseYou've already chosen the best response.1yea shure if its not too complicated

Rosh007
 2 years ago
Best ResponseYou've already chosen the best response.1well you are right ... now compare with your question......you got your answer

Rosh007
 2 years ago
Best ResponseYou've already chosen the best response.1that is you got \[\frac{ d }{ dx }(\arctan(x/a) = \frac{ a }{ x^2+a^2 }\]

malevolence19
 2 years ago
Best ResponseYou've already chosen the best response.0I 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) :/

Rosh007
 2 years ago
Best ResponseYou've already chosen the best response.1now itegrat on both sides.. you got answer

Rosh007
 2 years ago
Best ResponseYou've already chosen the best response.1Remember Integration is the opposite of differentiation @SABREEN .. now what will be tha answer?

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

Rosh007
 2 years ago
Best ResponseYou've already chosen the best response.1yes.. Welcome.. Now should I explain how 1/a comes ???? ;)

malevolence19
 2 years ago
Best ResponseYou've already chosen the best response.0Next one in line: \[\int\limits \tanh^{1}(ax)dx\]

SABREEN
 2 years ago
Best ResponseYou've already chosen the best response.1loll nahh i got this bro=)

Rosh007
 2 years ago
Best 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.. :)

malevolence19
 2 years ago
Best ResponseYou've already chosen the best response.0I 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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