Question

# Assertion :The value of ∫π0xf(sinx)dx is π2∫π0f(sinx)dx or π∫π/20f(sinx)dx Reason: ∫a0f(x)dx=∫a0f(a−x)dx and ∫2a0f(x)dx=2∫a0f(x)dx If f(2a−x)=f(x)

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Both Assertion and Reason are incorrect
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Solution

## The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for AssertionLet I=∫π0xf(sinx)dx =∫π0(π−x)f(sin(π−x))dx (∵∫a0f(x)dx=∫a0f(a−x)dx)⇒ 2I=π∫π0f(sinx)dx ⇒ I=π2∫π0f(sinx)dx =2.π2∫π20f(sinx)dx (∵f(sin(π−x))=f(sinx)) =π∫π20f(sinx)dx

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