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JEE Main 20179 AprilOnline
Exam
JEE Main
Year
2017
Date
9 April
Mode
Online
Subject
Mathematics
Chapter
Indefinite Integration

JEE Main 2017 · 9 April · Online · Mathematics · Indefinite Integration

If f((3x - 4)/(3x + 4)) = x + 2, x ≠ - (4)/(3) , and ∫ f(x)dx =...

If f(3x−43x+4)=x+2,x≠−43f\left(\dfrac{3x - 4}{3x + 4}\right) = x + 2, x \ne - \dfrac{4}{3}, and ∫f(x)dx=Alog∣1−x∣+Bx+C∫ f\left(x\right)dx = Alog\left|1 - x\right| + Bx + C , then the ordered pair \left(A, B\right) is equal to

Options

  1. A(−83,−23)\left(- \dfrac{8}{3}, - \dfrac{2}{3}\right)
  2. B(−83,23)\left(- \dfrac{8}{3}, \dfrac{2}{3}\right)
  3. C(83,23)\left(\dfrac{8}{3}, \dfrac{2}{3}\right)
  4. D(83,−23)\left(\dfrac{8}{3}, - \dfrac{2}{3}\right)

Correct answer

B. (−83,23)\left(- \dfrac{8}{3}, \dfrac{2}{3}\right)

Step-by-step solution

Given f(3x−43x+4)=x+2,x≠−43f\left(\dfrac{3x - 4}{3x + 4}\right) = x + 2, x \ne - \dfrac{4}{3}

Let, 3x−43x+4=t\dfrac{3x - 4}{3x + 4} = t

3x−4=3tx+4t3x - 4 = 3tx + 4t

x=4t+43−3t+2x = \dfrac{4t + 4}{3 - 3t} + 2

∴f(t)=10−2t3−3t∴ f\left(t\right) = \dfrac{10 - 2t}{3 - 3t}

⇒f(x)=2x−103x−3⇒ f\left(x\right) = \dfrac{2x - 10}{3x - 3}

∫f(x)dx=∫2x−103x−3dx∫ f\left(x\right)dx = ∫ \dfrac{2x - 10}{3x - 3}dx

=∫2x3x−3dx−10∫dx3x−3= ∫ \dfrac{2x}{3x - 3}dx - 10 ∫ \dfrac{dx}{3x - 3}

=2x3−83ln⁡∣x−1∣+C= \dfrac{2x}{3} - \dfrac{8}{3}\ln \left|x - 1\right| + C, where CC is the constant of integration.
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