1.

Let n in N such that n gt 1. Statement-1: int_(oo)^(0) (1)/(1+x^(n))dx=int_(0)^(1) (1)/((1-x^(n))^(1//n))dx Statement-2: int_a^b f(x)dx=int_(a)^(b) f(a+b-x)dx

Answer»

Statement-1 is true, Statement-2 is True,Statement-2 is a correct EXPLANATION for Statement-1.
Statement-1 is True, Statement-2 is not a correct explanation for Statement-1.
Statement-1 is True, Statement-2 is False.
Statement-1 is False, Statement-2 is True.

Solution :Clearly, statement-2 is true (See page 44.17 Property IX).
Putting `X^(N)=tan^(2)THETA`, we get
`I_(1)=underset(0)overset(oo)int (1)/(1+x^(n))DX=(2)/(n)underset(0)overset(pi//2)int tan^((2)/(n)-1)theta d theta`
Putting `x^(n)=sin^(2) theta`, we get
`I_(2)=(2)/(n)underset(0)overset(pi//2)int tan^((2)/(n)-1) theta d theta`
`:. underset(0)overset(oo)int(1)/(1+x^(n))dx=underset(0)overset(1)int(1)/((1-x^(n))^(1//n)dx`
So, statement-1 is also true.


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