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JEE Main 201510 AprilOnline
Exam
JEE Main
Year
2015
Date
10 April
Mode
Online
Subject
Mathematics
Chapter
Indefinite Integration

JEE Main 2015 · 10 April · Online · Mathematics · Indefinite Integration

The integral ∫ (dx)/((x + 1)(3)/(4)(x - 2)(5)/(4)) , is equal to

The integral ∫dx(x+1)34(x−2)54∫ \dfrac{dx}{\left(x + 1\right)\dfrac{3}{4}\left(x - 2\right)\dfrac{5}{4}}, is equal to

Options

  1. A4(x−2x+1)14+c4\left(\dfrac{x - 2}{x + 1}\right)\dfrac{1}{4} + c
  2. B−43(x+1x−2)14+c- \dfrac{4}{3}\left(\dfrac{x + 1}{x - 2}\right)\dfrac{1}{4} + c
  3. C4(x+1x−2)14+c4\left(\dfrac{x + 1}{x - 2}\right)\dfrac{1}{4} + c
  4. D−43(x−2x+1)14+c- \dfrac{4}{3}\left(\dfrac{x - 2}{x + 1}\right)\dfrac{1}{4} + c

Correct answer

B. −43(x+1x−2)14+c- \dfrac{4}{3}\left(\dfrac{x + 1}{x - 2}\right)\dfrac{1}{4} + c

Step-by-step solution

∫dx(x+1)34(x−2)54∫ \dfrac{dx}{\left(x + 1\right)\dfrac{3}{4}\left(x - 2\right)\dfrac{5}{4}}

=∫dx(x+1)2(x−2x+1)54= ∫ \dfrac{dx}{{\left(x + 1\right)}^{2}\left(\dfrac{x - 2}{x + 1}\right)\dfrac{5}{4}}

Put, x−2x+1=t\dfrac{x - 2}{x + 1} = t⇒3(x+1)2dx=dt⇒ \dfrac{3}{{\left(x + 1\right)}^{2}}dx = dt

∴I=−431t14+c∴ I = \dfrac{- 4}{3}\dfrac{1}{t\dfrac{1}{4}} + c

=−43(x+1x−2)14+c= \dfrac{- 4}{3}\left(\dfrac{x + 1}{x - 2}\right)\dfrac{1}{4} + c, cc being the constant of integration.
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