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Statement-1:int(0)^(1)(cos x)/(1+x^(2))d...

Statement-1:`int_(0)^(1)(cos x)/(1+x^(2))dxgt(pi)/(4)cos1`
Statement-2: If f(x) and g(x) are continuous on [a,b], then
`int_(a)^(b) f(x) g(x)dx=f(c )int_(a)^(b)g(x)` for some `c in (a,b)`.

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.

Text Solution

Verified by Experts

The correct Answer is:
A
Statement-2, being the statement of generalized mean value theorem, is true.
Using statement-2, these exists `c in(0,1)` such that `underset(0)overset(1)int (cosx)/(1+x^(2))dx=cos c underset(0)overset(1)int (1)/(1+x^(2))dx=(pi)/(4)cos c`
Clearly, `cos c gt cos 1` for all `c in (0,1)`
`rArr (pi)/(4)cos c gt (pi)/(4)cops 1`
`rArr underset(0)overset(1)int(cos x)/(1+x^(2))dx gt (pi)/(4)cos 1`
So, statement-1 is true. Also, statement-2 is a correct explanation for statement-1.
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