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  • Need explanation for: Which of the following is not always positive?

Which of the following is not always positive?

  • Option 1)

    \int_{a}^{b} |x|dx

  • Option 2)

    \int_{a}^{b} e^{-x^{2}} dx

  • Option 3)

    \int_{a}^{b}\ln x^{2} dx

  • Option 4)

    \int_{a}^{b}|\sin x|dx

 

Answers (1)

best_answer

As we have learnt,

 

Properties of Definite Integration -

If f(x)\geqslant 0 for all

x\in \left [ a,b \right ] then

\int_{a}^{b}f\left ( x \right )dx\geqslant 0

-

 

 \ln x^{2} < 0, \forall x\;\epsilon\;(0,1)

Thus,

\int_{a} ^{b}\ln x^{2}can be negetive if (a, b) are between 0 and 1.

 


Option 1)

\int_{a}^{b} |x|dx

Option 2)

\int_{a}^{b} e^{-x^{2}} dx

Option 3)

\int_{a}^{b}\ln x^{2} dx

Option 4)

\int_{a}^{b}|\sin x|dx

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Plabita

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