ExamHabit
Practise free

JEE Main 2015

JEE Main 2015 Mathematics question on Definite Integrals and Areas

The integral ∫24log⁡x2log⁡x2+log⁡(36−12x+x2)dx\int_2^4 \frac{\log x^2}{\log x^2 + \log(36 - 12x + x^2)} dx is equal to
  1. A.
    2
  2. B.
    4
  3. C.
    1
  4. D.
    6
Show answer and solution

Correct option: C

Solution

Note that log(36 - 12x + x²) = log((x-6)²) = 2log|x-6|. So the integral becomes ∫₂⁴ [2log x / (2log x + 2log(6-x))] dx = ∫₂⁴ [log x / (log x + log(6-x))] dx.

Using the property ∫ₐᵇ f(x)dx = ∫ₐᵇ f(a+b-x)dx with a+b=6, the substitution x → 6-x gives ∫₂⁴ [log(6-x) / (log(6-x) + log x)] dx.

Adding both forms: 2I = ∫₂⁴ 1 dx = 2, so I = 1.

[JEE Main 2015]

Practise Definite Integrals and Areas the way it is asked

A timed set of real JEE Main Definite Integrals and Areas questions, graded as the exam grades. No sign-up.

Start a Definite Integrals and Areas test

More: all JEE Main Definite Integrals and Areas questions · JEE Main 2015 · Mathematics chapter list