QJEEM13Mab89b: JEE Main 2013 Mathematics - Class 12 JEEM13Mab89b Solution: Statement-1 : The value of the integral is equal to Statement-2 :

Question

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Statement-1 : The value of the integral

\int\limits_{\pi /6}^{\pi /3} {{{dx} \over {1 + \sqrt {\tan \,x} }}}

is equal to

\pi /6

Statement-2 : $\int\limits_a^b {f\left( x \right)} dx = \int\limits_a^b {f\left( {a + b - x} \right)} dx.$

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 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.

Accepted Answer

Let

I = \int\limits_{\pi /6}^{\pi /3} {{{dx} \over {1 + \sqrt {\tan \,x} }}}

= \int\limits_{\pi /6}^{\pi /3} {{{dx} \over {\sqrt {\tan \left( {{\pi \over 2} - x} \right)} }}}

= \int\limits_{\pi /6}^{\pi /3} {{{\sqrt {\tan \,x} \,dx} \over {1 + \sqrt {\tan \,x} }}} \,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)

Also, Given,

I

= \int\limits_{\pi /6}^{\pi /3} {{{\sqrt {\tan \,x} \,dx} \over {1 + \sqrt {\tan \,x} }}} \,\,\,\,\,\,\,\,\,\,\,\,...\left( {ii} \right)

By adding

(1)

and

(2),

we get

2I = \int\limits_{\pi /6}^{\pi /3} {dx}

\Rightarrow I = {1 \over 2}\left[ {{\pi \over 3} - {\pi \over 6}} \right]

= {\pi \over {12}},

statements -

1

is false

\int\limits_a^b {f\left( x \right)dx = \int\limits_a^b {f\left( {a + b - x} \right)dx} }

It is fundamental property.

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