The value of the integral 

\int_{0}^{1}x\cot^{-1}(1-x^{2}+x^{4})dx is :

  • Option 1)

    \frac{\pi}{2}-\frac{1}{2}\:log_{e}\:2

  • Option 2)

    \frac{\pi}{4}-\:log_{e}\:2

  • Option 3)

    \frac{\pi}{2}-\:log_{e}\:2

  • Option 4)

    \frac{\pi}{4}-\frac{1}{2}\:log_{e}\:2

 

Answers (1)

\\\int_{0}^{1}x\cot^{-1}(1-x^{2}+x^{4})dx\\\\\\\:= \int_{0}^{1}x\:\tan^{-1}\left ( \frac{1}{1-x^{2}+x^{4}} \right )dx

put\:x^{2}=t

\\\frac{1}{2}\:\int_{0}^{1} \:\tan^{-1}\left ( \frac{1}{1-t+t^{2}} \right )dt\\\\\\\:=\int_{0}^{1}\frac{1}{2}\tan^{-1}\left ( \frac{t+(1+t)}{1-t(1-t)} \right )dt

\\\frac{1}{2}\:\int_{0}^{1} \:(tan^{-1}t+\tan^{-1}(1-t))dt\\\\\\:=\frac{1}{2}\int_{0}^{1}\tan^{-1}(1-t)dt+\frac{1}{2}\int_{0}^{1}\tan^{-1}(1-t)dt

\\\int_{0}^{1} \:\tan^{-1}t\:dt=\int_{0}^{1}\tan^{-1}(1-t)dt

put \:\:\tan^{-1}t=k

\\\int_{0}^{\pi/4}k\:\sec^{2}k\:dk\\\\\\\:=\pi/4-1/2\:ln \:\:2\:\:( using\:\:\:by\:\:parts)


Option 1)

\frac{\pi}{2}-\frac{1}{2}\:log_{e}\:2

Option 2)

\frac{\pi}{4}-\:log_{e}\:2

Option 3)

\frac{\pi}{2}-\:log_{e}\:2

Option 4)

\frac{\pi}{4}-\frac{1}{2}\:log_{e}\:2

Most Viewed Questions

Preparation Products

Knockout JEE Main (Six Month Subscription)

- AI Coach Study Modules, - Unlimited Mock Tests, - Study Improvement Plan.

₹ 9999/- ₹ 8499/-
Buy Now
Knockout JEE Main (Nine Month Subscription)

- AI Coach Study Modules, - Unlimited Mock Tests, - Study Improvement Plan.

₹ 13999/- ₹ 12499/-
Buy Now
Test Series JEE Main 2024

Chapter/Subject/Full Mock Tests for JEE Main, Personalized Performance Report, Weakness Sheet, Complete Answer Key,.

₹ 7999/- ₹ 4999/-
Buy Now
JEE Main Rank Booster 2023

Booster and Kadha Video Lectures, Unlimited Full Mock Test, Adaptive Time Table, Faculty Support.

₹ 9999/- ₹ 6999/-
Buy Now
Knockout JEE Main (One Month Subscription)

Personalized AI Tutor and Adaptive Time Table, Self Study Material, Unlimited Mock Tests and Personalized Analysis Reports, Faculty Support.

₹ 7999/- ₹ 4999/-
Buy Now
Boost your Preparation for JEE Main 2021 with Personlized Coaching
 
Exams
Articles
Questions