Integration -2

Calculus Level 4

If $\displaystyle I(a) = \int_0^\infty \dfrac{\tan^{-1} (ax) - \tan^{-1} x }x \, dx$ , then of which of the following is/are true?

Choose the most precise option.

(A): $I'(1) , I'(2), I'(3)$ follows a harmonic progression .
(B): $I'(2) = \dfrac \pi 4$ .
(C): $I(\pi) = \dfrac {\pi \ln \pi} 4$ .
(D): $I'(3) = \dfrac \pi 6$ .

C A,B,C A,B,C,D A,B,D

2 solutions

Spandan Senapati
Apr 23, 2017

This integral is popularly known as Frullani's Integral.Of the form of $\int f(bx)-f(ax)/x dx=(f(infinite)-f(0))ln(b/a)$ with the limits from $0-infinite$ .And for the function $f(infinite)$ and $f(0)$ exist.Like for the function $arctanx$ .So now its quite easy $I(a)=π/2lna$ .Thus $I'(a)=π/2a$ .So now its easy to find the values of $I'(1)=π/2,I'(2)=π/4,I'(3)=π/6$ .Clearly they form an $HP$ and $I(π)=π/2ln(π)$ .So $A,B,C,D$ are correct.The integral can be evaluated using Feynman's Trick(Special case of Newton-Leibnitz formula)

Ya...frullani is awesome!!!

A Former Brilliant Member - 4 years, 1 month ago

I got A,B,D. :P.. I didnt knew about the integral :(

Md Zuhair - 2 years, 8 months ago

Sanjoy Kundu
Jan 5, 2019

Trivial by Differentiation Under the Integral

Integration -2

2 solutions

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