Verified Solution • 2025–26 Session moderate

QJEEM07Ma7fb3: JEE Main 2007 Mathematics - Class 12 JEEM07Ma7fb3 Solution: equals

∫dxcos⁡x+3sin⁡x\int {{{dx} \over {\cos x + \sqrt 3 \sin x}}} equals

Multiple Choice Options

  • A log⁡ tan⁡ (x2+π12)+C\log \,\tan \,\left( {{x \over 2} + {\pi \over {12}}} \right) + C
  • B log⁡ tan⁡ (x2−π12)+C\log \,\tan \,\left( {{x \over 2} - {\pi \over {12}}} \right) + C
  • C  12 log⁡ tan⁡ (x2+π12)+C\,{1 \over 2}\,\log \,\tan \,\left( {{x \over 2} + {\pi \over {12}}} \right) + C
  • D  12 log⁡ tan⁡ (x2−π12)+C\,{1 \over 2}\,\log \,\tan \,\left( {{x \over 2} - {\pi \over {12}}} \right) + C

Step-by-Step Verified Solution

I=∫dxcos⁡x+3sin⁡xI = \int {{{dx} \over {\cos x + \sqrt 3 \sin x}}}

⇒I=∫dx2[12cos⁡x+32sin⁡x]\Rightarrow I = \int {{{dx} \over {2\left[ {{1 \over 2}\cos x + {{\sqrt 3 } \over 2}\sin x} \right]}}}

=12∫dx[sin⁡π6cos⁡x+cos⁡π6sin⁡x]= {1 \over 2}\int {{{dx} \over {\left[ {\sin {\pi \over 6}\cos x + \cos {\pi \over 6}\sin x} \right]}}}

=12.∫dxsin⁡(x+π6)= {1 \over 2}.\int {{{dx} \over {\sin \left( {x + {\pi \over 6}} \right)}}}

⇒I=12.∫cos⁡ec(x+π6)dx\Rightarrow I = {1 \over 2}.\int {\cos ec\left( {x + {\pi \over 6}} \right)dx}

But we know that

∫cos⁡ec x dx=log⁡∣(tan⁡x/2)∣+C\int {\cos ec\,x\,dx} = \log \left| {\left( {\tan x/2} \right)} \right| + C

∴\therefore I=12.log⁡ tan⁡(x2+π12)+CI = {1 \over 2}.\log \,\tan \left( {{x \over 2} + {\pi \over {12}}} \right) + C

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