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Let f(x)=(e^(x)+1)/(e^(-x)-1) and int(0)...

Let `f(x)=(e^(x)+1)/(e^(-x)-1) and int_(0)^(1)x^(3)(e^(x)+1)/(e^(x)-1) dx=alpha"then", int_(-1)^(1)int t^(3)f(t)dt` is equal to

A

0

B

`alpha`

C

`2alpha`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
Let `g(x)=x^(3) f(x)`. Then,
`g(-x)=(-x)^(3)f(-x)`
`rArr g(-x)-x^(3)((e^(-x)+1)/(e^(-x)-1))=-x^(3)((e^(x)+1)/(1-e^(x)))=x^(3)((e^(x)+1)/(e^(x)-1))=g(x)`
`rArr g(x)` is an even function.
Hence, `underset(-1)overset(1)intt^(3)f(t) dt=2underset(0)overset(1)int t^(3) f(t) dt=2 alpha`
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