Question 263 marksShort AnswerSection C

Evaluate: integral from 0 to 1 of xtan⁡−1x dxx \tan^{-1} x \, dx.

Integrals
Definite Integral by Parts

Official answer

The value of the definite integral is (π−2)/4(\pi - 2)/4.


Method: Using integration by parts with u=tan⁡−1xu = \tan^{-1}x and dv=x dxdv = x \, dx gives (x2/2)tan⁡−1x−12∫x2/(1+x2) dx(x^2/2)\tan^{-1}x - \tfrac{1}{2}\int x^2/(1+x^2) \, dx. Simplifying the remaining integral using x2/(1+x2)=1−1/(1+x2)x^2/(1+x^2) = 1 - 1/(1+x^2) and evaluating from 0 to 1 gives π/4−1/2\pi/4 - 1/2, i.e. (π−2)/4(\pi - 2)/4.

Answer keywords:integration by partsdefinite integraltan inverse xILATE rulelimits of integrationalgebraic and inverse trig function

Marking scheme

  1. 1.1 mark: correct choice of parts (u=tan⁡−1xu = \tan^{-1}x, dv=x dxdv = x \, dx) and correct du, v.
  2. 2.1 mark: correct simplification of x2/(1+x2)=1−1/(1+x2)x^2/(1+x^2) = 1 - 1/(1+x^2) and setting up the remaining integral.
  3. 3.1 mark: correct evaluation of limits and final answer (π−2)/4(\pi - 2)/4.

Hint

Take u=tan⁡−1xu = \tan^{-1}x, dv=x dxdv = x \, dx; use integration by parts, then simplify x2/(1+x2)=1−1/(1+x2)x^2/(1+x^2) = 1 - 1/(1+x^2).

Quick oral answer

Using integration by parts with u=tan⁡−1xu = \tan^{-1}x and v=x2/2v = x^2/2, the integral evaluates to (π−2)/4(\pi - 2)/4 after simplifying x2/(1+x2)x^2/(1+x^2).

Common mistakes

  1. 1.Choosing x as the first function (u) instead of tan⁡−1x\tan^{-1}x, violating the ILATE priority and complicating the remaining integral.
  2. 2.Forgetting to rewrite x2/(1+x2)x^2/(1+x^2) as 1−1/(1+x2)1 - 1/(1+x^2) before integrating, leaving the integral unresolved.
  3. 3.Sign errors while substituting the limits 0 and 1 into the tan⁡−1x\tan^{-1}x and x terms.

Analysis and explanation

This is a classic ILATE-rule integration-by-parts question combined with definite-integral evaluation.


Concept: Since tan⁡−1x\tan^{-1}x is inverse trigonometric and xx is algebraic, ILATE priority makes tan⁡−1x\tan^{-1}x the 'first function' (uu) and x dxx \, dx the 'second function' (dvdv) — this ordering keeps the resulting integral simple.


Exam trap: A common error is choosing u = x instead of u = tan⁻¹x, or forgetting to simplify x2/(1+x2)x^2/(1+x^2) before integrating, leaving an unresolved rational integral.


Real-world link: Definite integrals of this form appear in computing weighted-average phase angles and moment calculations in engineering problems involving arctangent-weighted distributions.

Interesting facts

Integration by parts is derived directly from the product rule of differentiation, reversed and rearranged.

Definite integrals involving tan⁻¹x often produce results containing π, linking algebraic areas to circular geometry.

Question details

Q26 of 44 questions

Board
CBSE
Class
Class 12
Subject
Mathematics
Year
2026
Section
Section C
Marks
3
Type
Short Answer
Chapter
Integrals
Topic
Definite Integral by Parts
Bloom's level
apply
Syllabus topic
Unit 4: Integrals — Definite Integrals using Integration by Parts
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

Why is tan⁡−1x\tan^{-1}x chosen as the first function in integration by parts?

By the ILATE priority rule (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential), inverse trigonometric functions are taken as u before algebraic functions, since their derivative is algebraic and simplifies the resulting integral.

What is the final numeric value of the integral?

It equals (π−2)/4(\pi - 2)/4, approximately 0.2854.