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Please solve rd sharma class 12 chapter Indefinite integrals exercise 18.4 question 6 maths textbook solution

Answers (1)

Answer:

$2\log \left | x-1) \right |-\frac{1}{(x-1)}+C$

Hint:

Use integration by partial fraction.

Given:

$\int \frac{2x-1}{\left ( x-1 \right )^{2}}dx$

Solution:

$\frac{2x-1}{\left ( x-1 \right )^{2}}$

$=\frac{2(x-1)+1}{\left ( x-1 \right )^{2}}$

$=\frac{2}{x-1}+\frac{1}{(x-1)^{2}}$

$\Rightarrow \int \frac{2x-1}{\left ( x-1 \right )^{2}}dx=\int \left [ \frac{2}{x-1} +\frac{1}{(x-1)^{2}}\right ]dx$

$=2\int \frac{1}{x+1}dx+\int \frac{1}{(x-1)^{2}}dx$

$=2\log \left | x-1) \right |-\frac{1}{(x-1)}+C$

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