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The value of int(-pi//2)^(pi//2)(x^(2)co...

The value of `int_(-pi//2)^(pi//2)(x^(2)cosx)/(1+e^(x))` d x is equal to

A

`(pi^(2))/(4)-2`

B

`(pi^(2))/(4)+2`

C

`pi^(2)-e^(-pi//2)`

D

`pi^(2)-e+^(pi//2)`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `I=int_(-pi//2)^(pi//2)(x^(2)cosx)/(1+e^(x))dx` . . . (i)
`[:'int_(a)^(b)f (x)dx=int_(a)^(b)f(a+b-x)dx]`
`rArrI=int_(-pi//2)^(pi//2)(x^(2)cos(-x))/(1+e^(-x))` . . . (ii)
On adding Eqs . (i) and (ii) , we get
`2I=int_(-pi//2)^(pi//2)x^(2)cosx[(1)/(1+e^(x))+(1)/(1+e^(-x))]dx`
`=int_(-pi//2)^(pi//2)cos x* (1)dx`
`[:'int_(-a)^(a)f(x)dx=2int_(0)^(a)f(x)dx, "when" f (-x)=f(x)]`
`rArr2I=2int_(0)^(pi//2)x^(2)cosx dx`
Using integration by parts , we get
`2I=2[x^(2)(sinx)-(2x)(-cosx)+(2)(-sinx)]_(0)^(pi//2)`
`rArr2I=2[(pi^(2))/(4)-2]`
`:.I=(pi^(2))/(4)-2`
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