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Evaluate \int \frac{2x-3}{x^{3}+x}dx

  • Option 1)

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

  • Option 2)

    -3\ln \left | x \right |+\frac{3}{2}\ln \left | x^{2}+1 \right |+2\tan ^{-1}x+C

  • Option 3)

    -3\ln \left | x \right |-\frac{3}{2}\ln \left | x^{2}+1 \right |+2\tan ^{-1}x+C

  • Option 4)

    3\ln \left | x \right |+\frac{3}{2}\ln \left | x^{2}+1 \right |-2\tan ^{-1}x+C

 

Answers (1)

best_answer

As we learned,

 

Rule for Partial fraction -

Quadratic and Non-repeated.

Let Q(x)=(x-\alpha _{1})(x-\alpha _{2})\cdot \cdot \cdot \cdot (ax^{2}+bx+c)

Then,

\frac{P(x)}{Q(x)}=\frac{Ax+B}{ax^{2}+bx+c}+\frac{c}{x-\alpha _{1}}+\frac{D}{x-\alpha _{2}}+\cdot \cdot \cdot

- wherein

Find A,B,C\cdot \cdot \cdot

By comparing with P(x)

 

 

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

=-3\ln \left | x \right |+\frac{3}{2}\int \frac{2xdx}{x^{2}+1}+2\int \frac{dx}{x^{2}+1}

=-3\ln \left | x \right |+\frac{3}{2}\ln \left | x^{2}+1 \right |+2\tan ^{-1}x+C


Option 1)

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

Option 2)

-3\ln \left | x \right |+\frac{3}{2}\ln \left | x^{2}+1 \right |+2\tan ^{-1}x+C

Option 3)

-3\ln \left | x \right |-\frac{3}{2}\ln \left | x^{2}+1 \right |+2\tan ^{-1}x+C

Option 4)

3\ln \left | x \right |+\frac{3}{2}\ln \left | x^{2}+1 \right |-2\tan ^{-1}x+C

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gaurav

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