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\int_{-1}^{1}\frac{log\left ( x+\sqrt{1+x^{2}} \right )}{x+log\left ( x+\sqrt{1+x^{2}} \right )}\left ( f(x)-f(-x) \right )dx  is euqal to?

  • Option 1)

    0

  • Option 2)

    2\int_{0}^{1}\frac{log\left ( x+\sqrt{1+x^{2}} \right )}{x+log\left ( x+\sqrt{1+x^{2}} \right )}\left ( f(x)-f(-x) \right )dx

  • Option 3)

    2f(x)

  • Option 4)

    none of these

 

Answers (1)

best_answer

As we learnt

Properties of definite integration -

If f\left ( x \right ) is an EVEN function of x: then integral of the function from - a to a is the same as twice the integral of the same function from o to a.

\int_{-a}^{a}f(x)dx= 2\left \{ \int_{o}^{a} f(x)dx\right \}

 

- wherein

Check even function f(-x)=f(x) and symmetrical about y axis.

 

 Let\phi (x)=\frac{log\left ( x+\sqrt{1+x^{2}} \right )}{x+log\left ( x+\sqrt{1+x^{2}} \right )};g(x)=\left ( f(x)-f(-x) \right )

f(x) is an even function Þ \int_{-1}^{1}(\phi (x).g(x))dx=0

            g(x) is an odd function.


Option 1)

0

Option 2)

2\int_{0}^{1}\frac{log\left ( x+\sqrt{1+x^{2}} \right )}{x+log\left ( x+\sqrt{1+x^{2}} \right )}\left ( f(x)-f(-x) \right )dx

Option 3)

2f(x)

Option 4)

none of these

Posted by

Aadil

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