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Question

# Assertion :If n>1 then Statement -1: ∫∞0dx1+xn=∫10dx(1−xn)1/n Reason: Statement -2: ∫baf(x)dx=∫baf(a+b−x)dx

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 B Both Assertion and Reason are correct but Reason is not the correct explanation for AssertionPut xn=tan2θ⇒nxn−1dx=2tanθsec2θdθ⇒dx=2tanθsec2θ2tan2(n−1)/nθdθ∫∞0dx1+xn=2n∫π/20tan1−2+2/nθdθ=2n∫π/20tan2/n−1θdθPutting xn=sin2θ in R.H.S. of statement 1⇒nxn−1dx=2sinθcosθdθ⇒dx=2sinθcosθnsin(n−1)/nθ=∫10dx(1−xn)1/n=2n∫π/201(cos2/nθ)sin2/n−1θcosθdθ=2n∫π/20tan2/n−1θdθClearly statement 2 is true

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