Statement I: The value of the integral `int_(pi//6)^(pi//3) (dx)/(1+sqrt(tanx))` is equal to `(pi)/6`.
Statement II: `int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)dx`

Options

A. Statement I is correct , Statement II is correct ,
Statement I is a correct explanation for Statement I
B. Statement I is correct , Statement II is correct,
Statement II is not a correct explanation for Statement I
C. Statement I is correct , Statement II is false
D. Statement I is incorrect , Statement II is correct

Answer

D

Explanation

Let `I=int_(pi//6)^(pi//3)(dx)/(1+sqrt(tanx))` . . . (i)
`:.I=int_(pi//6)^(pi//3)(dx)/(1+sqrt(tan((pi)/(2)-x)))`
` =int_(pi//6)^(pi//3)(dx)/(1+sqrt(cotx))`
`rArrI=int_(pi//6)^(pi//3)(sqrt(tanx)dx)/(1+sqrt(tanx))` . . . (ii)
On adding Eqs . (i) and (ii) , we get
`2I=int_(pi//6)^(pi//3)dx`
`rArr2I=[x]_(pi//6)^(pi//3)dx`
`rArrI=(1)/(2)[(pi)/(3)-(pi)/(6)]=(pi)/(12)` Statement I is false .
But `int_(a)^(b)f(x)dx=int_(a)^(b)f(a+b-x)` dx is a true statement by property of definite integrals.