Find \int \frac{x+1}{x(1-2 x)} \dx

CBSE Class 12 Math 2020 Delhi Set 2 Solved Paper

Question : 7 of 11

Marks: +1, -0

Find

∫ \;{x+1}/{x(1-2 x)} \dx

Solution:

Let

I=∫ \;{x+1}/{x(1-2 x)} \dx

Also, let

\;\;\text " Also, let "\;\;\;{x+1}/{x(1-2 x)}\;=\;{ {\A}}/{x}+\;{ {\B}}/{1-2 x}

⇒ \;{x+1}/{x(1-2 x)}\;=\;{ {\A}(1-2 x)+ {\B} x}/{x(1-2 x)}

⇒\;x+1\;=x(-2 {\A}+ {\B})+ {\A}

Comparing coefficient of

x

both sides, we get

1=-2 A+B

Comparing constant term both sides, we get

{\A}=1

∴ \;\;

From equation (i),

1=-2(1)+B

⇒ \;\; {\B}=3

∴ \;\; ∫ \;{x+1}/{x(1-2 x)} \dx=∫ \;{1}/{x} \dx+∫ \;{3}/{1-2 x} \dx

=\log |x|+\;{3 \log |1-2 x|}/{-2}+C

=\log |x|-\;{3}/{2} \log |1-2 x|+C \;\text " Ans. "\;

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