## SABREEN Group Title 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

1. VeritasVosLiberabit Group Title

$\int\limits_{}^{}\frac{ z }{ x ^{2}+z ^{2} }dx$

2. VeritasVosLiberabit Group Title

this is a formula you use to compute the integral of a function with this structure.

3. Rosh007 Group Title

You will be able to relate your question simply to the standard integral $\large \frac{1 }{ a^2+x^2 } =\frac{1 }{a } \times \arctan (\frac{x}{ a })$

4. Rosh007 Group Title

if Left hand side is the function that has to integrate with respect to x

5. SABREEN Group Title

is 1/a the derivative of x/a

6. SABREEN Group Title

is tht why its there?

7. VeritasVosLiberabit Group Title

1/a is part of the standard formula. I'll work it out for you.

8. VeritasVosLiberabit Group Title

$z \int\limits_{}^{}\frac{ 1 }{ x ^{2}+z ^{2} }dx$

9. SABREEN Group Title

thnk you id like to see why

10. VeritasVosLiberabit Group Title

$z*\frac{ 1 }{ z }\arctan(\frac{ x }{ z })\rightarrow \arctan(\frac{ x }{ z })$

11. VeritasVosLiberabit Group Title

in this case z is a scalar.

12. VeritasVosLiberabit Group Title

so typically z would be represented as a number. I think they are using zahlen which denotes integer values.

13. SABREEN Group Title

why is arc tan (x/z) and where does the 1/z come from??

14. VeritasVosLiberabit Group Title

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.

15. Rosh007 Group Title

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 ??

16. SABREEN Group Title

it is 1/1+x^2

17. Rosh007 Group Title

well,good.. then what will be derivative of arctan(x/a)

18. SABREEN Group Title

i dunno

19. Rosh007 Group Title

20. Rosh007 Group Title

and rember x/a is another function so you need to apply the chain rule of differentiation

21. SABREEN Group Title

1/(1+(x/a)^2)*(1/a)

22. Rosh007 Group Title

Yup........... You got it.. simplify it

23. malevolence19 Group Title

I mean, I can do the integral if you want it in all the gruesome detail...

24. SABREEN Group Title

a/(x2+a^2)

25. SABREEN Group Title

yea shure if its not too complicated

26. Rosh007 Group Title

27. Rosh007 Group Title

that is you got $\frac{ d }{ dx }(\arctan(x/a) = \frac{ a }{ x^2+a^2 }$

28. malevolence19 Group Title

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) :/

29. Rosh007 Group Title

now itegrat on both sides.. you got answer

30. Rosh007 Group Title

Remember Integration is the opposite of differentiation @SABREEN .. now what will be tha answer?

31. SABREEN Group Title

so than the integral of a/(x^2+a^2) is arctan (x/a)

32. SABREEN Group Title

thanks=)

33. Rosh007 Group Title

yes.. Welcome.. Now should I explain how 1/a comes ???? ;)

34. malevolence19 Group Title

Next one in line: $\int\limits \tanh^{-1}(ax)dx$

35. SABREEN Group Title

loll nahh i got this bro=)

36. Rosh007 Group Title

@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.. :)

37. malevolence19 Group Title

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)