Verified Solution • 2025–26 Session moderate

QJEEM08Maecb1: JEE Main 2008 Mathematics - Class 12 JEEM08Maecb1 Solution

The value of

\sqrt 2 \int {{{\sin xdx} \over {\sin \left( {x - {\pi \over 4}} \right)}}}

is

Multiple Choice Options

A $\,x + \log \,\left| {\,\cos \left( {x - {\pi \over 4}} \right)\,} \right| + c$
B $\,x - \log \,\left| {\,\sin \left( {x - {\pi \over 4}} \right)\,} \right| + c$
C $\,x + \log \,\left| {\,\sin \left( {x - {\pi \over 4}} \right)\,} \right| + c$
D $\,x - \log \,\left| {\,\cos \left( {x - {\pi \over 4}} \right)\,} \right| + c$

Step-by-Step Verified Solution

Let

I = \sqrt 2 \int {{{\sin \,xdx} \over {\sin \left( {x - {\pi \over 4}} \right)}}}

Put

x - {\pi \over 4} = t

\Rightarrow dx = dt

\Rightarrow I = \sqrt 2 \int {{{\sin \left( {t + {\pi \over 4}} \right)} \over {\sin \,t}}} dt

\,\,\,\,\,\,\,\,\,\,\,\, = {{\sqrt 2 } \over {\sqrt 2 }}\int {\left( {{{\sin t + \cos t} \over {\sin t}}} \right)} \,\,dt

\Rightarrow I = \int {\left( {1 + \cot \,t} \right)} dt

\,\,\,\,\,\,\,\,\,\,\,\, = t + \log \left| {\sin t} \right| + {c_1}

= x - {\pi \over 4} + \log \left| {\sin \left( {x - {\pi \over 4}} \right)} \right| + {c_1}

= x + \log \left| {\sin \left( {x - {\pi \over 4}} \right)} \right| + c

\left( \, \right.

where

{c = {c_1} - {\pi \over 4}}

\left. \, \right)

Related Learning Resources & Exam Preparation

JEE Main PYQ Question Bank

Browse year-wise authentic previous years' questions with solutions

Authentic CBT Mock Tests

Simulate official exam environment with live timer and percentile scoring