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\int \frac{(x+3)}{(x-1)^{3}}dx

  • Option 1)

    \frac{1}{(x-1)^{2}}+\frac{2}{(x-1)^{3}}+C

  • Option 2)

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

  • Option 3)

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

  • Option 4)

    -\frac{1}{(x-1)^{2}}-\frac{2}{(x-1)^{3}}+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{(x+3)}{(x-1)^{3}} dx=\int \frac{A}{(x-1)}+\frac{B}{(x-1^{2})}+\frac{C}{(x-1)^{3}}dx

On solving A= 0 ; B= 1 ; C= 4

Thus I= \int \frac{1}{(x-1^{2})}-\frac{4}{(x-1^{2})}= \frac{-(x+1)}{(x-1)^{2}}+ C

 

 


Option 1)

\frac{1}{(x-1)^{2}}+\frac{2}{(x-1)^{3}}+C

This is incorrect

Option 2)

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

This is incorrect

Option 3)

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

This is correct

Option 4)

-\frac{1}{(x-1)^{2}}-\frac{2}{(x-1)^{3}}+C

This is incorrect

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Aadil

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