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The integral

\int_{2}^{4}\frac{\log x^{2}}{\log x^{2}+\log (36-12x+x^{2})}dx   is equal to:

  • Option 1)

    2

  • Option 2)

    4

  • Option 3)

    1

  • Option 4)

    6

 

Answers (1)

best_answer

As learnt in concept

Properties of Definite integration -

\int_{a}^{b}f\left ( x \right )dx= \int_{a}^{b}f\left ( a+b-x \right )dx

When \int_{0}^{b}f\left ( x \right )dx= \int_{0}^{b}f\left ( b-x \right )dx

 

- wherein

Put the \left ( a+b-x \right ) at the place of x in f\left ( x \right )

 

I=\int_{2}^{4}\frac{logx^{2}dx}{logx^{2}+log(36-12x+x^{2})}

I=\int_{2}^{4}\frac{logxdx}{logx+log(6-x)}

Also,

\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx

I=\int_{2}^{4}\frac{log(6-x)dx}{logx+log(6-x)}

2I=\int_{2}^{4}dx=[x]_{2}^{4}

I=1


Option 1)

2

this option is incorrect

Option 2)

4

this option is incorrect

Option 3)

1

this option is correct

Option 4)

6

this option is incorrect

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