If the integral ∫ (5 x)/( x-2) d x=x+a | x-2 x|+k , then a is equal to

Correct answer: D. 2 ∫ (5 x)/( x-2) d x=∫ (5 x)/( x-2 x) d x ∫[(2( x+2 x)+( x-2 x))/( x-2 x)] d x =2 ∫(( x+2 x)/( x-2 x)) d x+∫ d x+k =2 | x-2 x|+x+k a=2

JEE Main 2012Offline

Exam: JEE Main
Year: 2012
Mode: Offline
Subject: Mathematics
Chapter: Indefinite Integration

JEE Main 2012 · Offline · Mathematics · Indefinite Integration

If the integral ∫ (5 x)/( x-2) d x=x+a | x-2 x|+k , then a is equal...

If the integral

\int \frac{5 \tan x}{\tan x-2} d x=x+a \ln |\sin x-2 \cos x|+k

, then

a

is equal to

Options

A $-1$
B $-2$
C $1$
D $2$

Correct answer

D. $2$

Step-by-step solution

\int \frac{5 \tan x}{\tan x-2} d x=\int \frac{5 \sin x}{\sin x-2 \cos x} d x

\Rightarrow \int\left[\frac{2(\cos x+2 \sin x)+(\sin x-2 \cos x)}{\sin x-2 \cos x}\right] d x

=2 \int\left(\frac{\cos x+2 \sin x}{\sin x-2 \cos x}\right) d x+\int d x+k

=2 \log |\sin x-2 \cos x|+x+k \quad \therefore a=2

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