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\int \frac{2x-1}{(x-5)^{2}}dx

  • Option 1)

    2ln |x-5|+\frac{9}{(x-5)}+C

  • Option 2)

    -2ln |x-5|+\frac{9}{(x-5)}+C

  • Option 3)

    2ln |x-5|-\frac{9}{(x-5)}+C

  • Option 4)

    -2ln |x-5|-\frac{9}{(x-5)}+C

 

Answers (1)

best_answer

As we have learned

Rule for Partial fraction -

Linear and repeated :

\frac{P(x)}{Q(x)}=\frac{P(x)}{(x-a)^{k}(x-a_{1})(x-a_{2})\cdot \cdot \cdot }

\frac{P(x)}{Q(x)}=\frac{A_{1}}{(x-a)}+\frac{A_{2}}{(x-a)^{2}}+\cdot \cdot \cdot \frac{A_{k}}{(x-a)^k}+\frac{A_{k+1}}{(x-a_{1})}+\frac{A_{k+2}}{(x-a_{2})}\cdot \cdot \cdot

- wherein

Where k>1

 

Where find

A_{1} , A_{2} ,A_{3}

by comparing with P(x)

 

 

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

2x-1= A(x-5)+B

On solving A=2, B=9

Thus \int \frac{2x-1}{(x-5)^{2}}dx = \int \frac{2dx}{(x-5)}+ \int \frac{9}{(x-5)^{2}}dx

\Rightarrow 2ln \left | (x-5) \right |- \frac{9}{(x-5)}+ C

 

 

 

 

 

 


Option 1)

2ln |x-5|+\frac{9}{(x-5)}+C

This is incorrect

Option 2)

-2ln |x-5|+\frac{9}{(x-5)}+C

This is incorrect

Option 3)

2ln |x-5|-\frac{9}{(x-5)}+C

This is correct

Option 4)

-2ln |x-5|-\frac{9}{(x-5)}+C

This is incorrect

Posted by

Aadil

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