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`

Options

A. Statement-1 is true, Statement-2 is True,Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer

B

Explanation

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.