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need solution for rd sharma maths class 12 chapter Indefinite integrals exercise 18.4 question 4

Answers (1)

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Answer:

$2\log \left | x-1 \right |-\frac{5}{x-1}+C$

Hint:

Use integration by partial fraction.

Given:

$\int \frac{2x+3}{\left ( x-1 \right )^{2}}dx$

Solution:

$\frac{2x+3}{\left ( x-1 \right )^{2}}$

$=\frac{2x-2+2+3}{\left ( x-1 \right )^{2}}$

$=\frac{2\left ( x-1 \right )+5}{\left ( x-1 \right )^{2}}$

$=\frac{2}{\left ( x-1 \right )}+\frac{5}{\left ( x-1 \right )^{2}}$

$I=\int \frac{2}{x-1}dx+\int \frac{5}{(x-1)^{2}}dx$

$\left [ \int \frac{1}{x}dx= \log \left | x \right |+C \right ]=2$

$I= \log \left | x-1 \right |-\frac{5}{x-1}+C$

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