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JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Critical Thinking

  • question_answer
    If \[\int_{{}}^{{}}{(\sin 2x+\cos 2x)\ dx=\frac{1}{\sqrt{2}}\sin (2x-c)+a}\], then the value of a and c is             [Roorkee 1978]

    A) \[c=\pi /4\] and \[a=k\] (an arbitrary constant)

    B) \[c=-\pi /4\] and \[a=\pi /2\]

    C) \[c=\pi /2\] and a is an arbitrary constant           

    D) None of these

    Correct Answer: A

    Solution :

    • \[\int_{{}}^{{}}{(\sin 2x+\cos 2x)\,dx}=-\frac{\cos 2x}{2}+\frac{\sin 2x}{2}+k\]                   
    • \[=\frac{1}{\sqrt{2}}\left( \sin 2x\cos \frac{\pi }{4}-\cos 2x\sin \frac{\pi }{4} \right)+k\]                   
    • \[=\frac{1}{\sqrt{2}}\sin \left( 2x-\frac{\pi }{4} \right)+k\]                
    • \[\Rightarrow c=\frac{\pi }{4}\] and \[a=k,\] an arbitrary constant.


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