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If  \int_{0}^{\pi }x\; f(sinx)dx=A\int_{0}^{\pi /2}f(sinx)dx,then \; A\; \; is

  • Option 1)

    \pi /4\;

  • Option 2)

    \; \; \; \pi \;

  • Option 3)

    \; \; 0\; \;

  • Option 4)

    \; 2\pi

 

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_{o}^{\pi }xf(sinx) dx

According to property \int_{0}^{a}f(x)dx=\int_{o}^{a}f(a-x)dx

I=\int_{0}^{\pi }(\pi -x)f(sin(\pi -x)dx

I=\int_{0}^{\pi }(\pi -x)f(sinx)dx

2I=\pi \int_{0}^{\pi }f(sinx)dx=2\pi \int_{0}^{\frac{\pi }{2}}f(sinx)dx

I=\pi \int_{0}^{\frac{\pi }{2}}f(sinx) dx


Option 1)

\pi /4\;

This option is incorrect

Option 2)

\; \; \; \pi \;

This option is correct

Option 3)

\; \; 0\; \;

This option is incorrect

Option 4)

\; 2\pi

This option is incorrect

Posted by

prateek

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