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JEE Main 201212 MayOnline

Exam: JEE Main
Year: 2012
Date: 12 May
Mode: Online
Subject: Mathematics
Chapter: Indefinite Integration

JEE Main 2012 · 12 May · Online · Mathematics · Indefinite Integration

The integral of (x^2-x)/(x^3-x^2+x-1) w.r.t. x is

The integral of

\frac{x^2-x}{x^3-x^2+x-1}

w.r.t.

x

is

Options

A $\frac{1}{2} \log \left(x^2+1+c\right)$
B $\frac{1}{2} \log \left|x^2-1\right|+c$
C $\log \left(x^2+1+c\right)$
D $\log \left|x^2-1\right|+c$

Correct answer

A. $\frac{1}{2} \log \left(x^2+1+c\right)$

Step-by-step solution

Let

I=\int \frac{x^2-x}{x^3-x^2+x-1} d x=\int \frac{x(x-1)}{x^2(x-1)+(x-1)}

d x=

\int \frac{x d x}{x^2+1} \left.=\frac{1}{2} \int \frac{2 x d x}{ \left(x^2+1\right.}\right)Let

x^2+1=t \Rightarrow 2 x d x=d t

\begin{aligned} & \therefore I=\frac{1}{2} \int \frac{d t}{t}=\frac{1}{2} \logt+c \\ & =\frac{1}{2} \log \left(x^2+1\right)+c \end{aligned}

where '

c

' is the constant of integration.

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