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  • Need clarity, kindly explain! - Integral Calculus - JEE Main-10

Integrate \frac{3x+1}{x^{2}(x^{2}+25)}  with respect to x

  • Option 1)

    \frac{1}{25}(3ln |x|- \frac{1}{x}+ 3/2ln|x^{2}+25|-1/5 \tan ^{-1}x)

  • Option 2)

    \frac{1}{25}(3ln |x|+ \frac{1}{x}- 3/2ln|x^{2}+25|+1/5 \tan ^{-1}x)

  • Option 3)

    \frac{1}{25}(3ln |x|- \frac{1}{x}- 3/2ln|x^{2}+25|-1/5 \tan ^{-1}x)

  • Option 4)

    \frac{1}{25}(3ln |x|+\frac{1}{x}+ 3/2ln|x^{2}+25|-1/5 \tan ^{-1}x)

 

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{(3x+1)dx}{x^{2}(x^{2}+25)}=\int \frac{A}{x}+\frac{B}{x^{2}}+\frac{Cx+D}{x^{2}+25}dx

\Rightarrow A=3/25; B=1/25; C = -3/25 , D= -1/25 

 

 

 

 


Option 1)

\frac{1}{25}(3ln |x|- \frac{1}{x}+ 3/2ln|x^{2}+25|-1/5 \tan ^{-1}x)

This is incorrect

Option 2)

\frac{1}{25}(3ln |x|+ \frac{1}{x}- 3/2ln|x^{2}+25|+1/5 \tan ^{-1}x)

This is incorrect

Option 3)

\frac{1}{25}(3ln |x|- \frac{1}{x}- 3/2ln|x^{2}+25|-1/5 \tan ^{-1}x)

This is correct

Option 4)

\frac{1}{25}(3ln |x|+\frac{1}{x}+ 3/2ln|x^{2}+25|-1/5 \tan ^{-1}x)

This is incorrect

Posted by

prateek

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