Question

A fun integral question:

Answer 1

\[\LARGE \int\limits_0^\infty \frac{\tan^{-1}(ax)-\tan^{-1}(x)}{x}dx\]

Answer 2

I am thinking
Differentiation Under Integral Sign Technique

Answer 3

\(\Large \dfrac{\partial I}{\partial a} = \int \limits_0^\infty \dfrac{1}{a^2 x^2+1 }dx\)

Answer 4

I = pi/2 ln a

Answer 5

Interesting! I had a different path to the solution in mind. \[\Large \int\limits_0^\infty \int\limits _x^{ax}\frac{1}{x(1+y^2)}dydx\] Reverse the order of integration:\[\Large \int\limits_0^\infty \int\limits _{y/a}^y \frac{1}{x(1+y^2)}dxdy = \int\limits_0^\infty \frac{lny-lny+lna}{1+y^2}dy \] \[\Large \ln a \frac{\pi}{2}\]

Same answer, but kind of like opposite methods haha.

Answer 6

I am checking if Laplace Transform helps to find us an alternate way :)

Answer 7

Interesting kainui,
can you brief on how you got that first double integral ?

Answer 8

y = arctan x ?

Answer 9

Yeah, try evaluating the integral for dy to get the original integral back.

Answer 10

that path is easy, but reverse part is quite difficult

Answer 11

Since every integral ends with a function evaluated at two points and them subtracted, I just sort of decided to try that.

Answer 12

@SithsAndGiggles and @ganeshie8 might also find this interesting :)

Answer 13

Similarly you can make a messed up definition of the derivative if you do something fun like this haha, \[\lim_{h \to 0} \int\limits_{f(x)} ^{f(x+h)} \frac{dx}{h}= f'(x)\]

But at anyway, I can try to give more explanation or reasoning if I didn't really explain what you were asking, I'm not sure I answered your question.

Answer 14

you did, clearly :)

Answer 15

@hartnn If you know any good problems to solve with differentiation under the integral sign I'd like to see one since I don't get enough practice with that since it's usually pretty hard to see when to use it.

Answer 16

I generally think of DUIS when there is one constant in a complicated looking function to be integrated.

Answer 17

I did one DUIS problem with ganeshie few days ago here, but my profile isn't loading that many questions of the past... .
I'll try to find one from other site..

Answer 18

wiki examples are quite good...
http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign

Answer 19

I've yet to get a grasp on the intuition/strategic mindset is DUIS... Any good resources or tutorials for that?