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Evaluate \int \frac{2x-3}{x^{3}+x}dx

  • Option 1)

    3ln |x|+ 3/2 ln |(x^{2}+1)|- 2 \tan ^{-1}x + C

  • Option 2)

    -3ln |x|+ 3/2 ln |(x^{2}+1)|+ 2 \tan ^{-1}x + C

  • Option 3)

    -3ln |x|- 3/2 ln |(x^{2}+1)|+ 2 \tan ^{-1}x + C

  • Option 4)

    3ln |x|+ 3/2 ln |(x^{2}+1)|- 2 \tan ^{-1}x + 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-3}{x^{3}+x}dx=\int \frac{2x-3}{x(x^{2}+1)}dx=\int \frac{A}{x}+\frac{Bx+C}{x^{2}+1}

A=-3; B =3  ; C= 2

Thus \int \frac{2x-3}{x^{3}+x}= \int \frac{-3}{x}dx + \int \frac{3x+2}{x^{2}+1}dx

-3ln |x|+ 3/2\int \frac{2x}{x^{2}+1}dx+2\int \frac{dx}{x^{2}+1}

=-3ln |x|+ 3/2 ln |x^{2}+1|+ 2 \tan ^{-1}x + C

 

 

 

 


Option 1)

3ln |x|+ 3/2 ln |(x^{2}+1)|- 2 \tan ^{-1}x + C

This is incorrect

Option 2)

-3ln |x|+ 3/2 ln |(x^{2}+1)|+ 2 \tan ^{-1}x + C

This is correct

Option 3)

-3ln |x|- 3/2 ln |(x^{2}+1)|+ 2 \tan ^{-1}x + C

This is incorrect

Option 4)

3ln |x|+ 3/2 ln |(x^{2}+1)|- 2 \tan ^{-1}x + C

This is incorrect

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gaurav

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