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JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Integration by Substitution

  • question_answer
    \[\int_{{}}^{{}}{\frac{\cos 2x+x+1}{{{x}^{2}}+\sin 2x+2x}}\ dx=\] [AI CBSE 1980]

    A)            \[\log ({{x}^{2}}+\sin 2x+2x)+c\]     

    B)            \[-\log ({{x}^{2}}+\sin 2x+2x)+c\]

    C)            \[\frac{1}{2}\log ({{x}^{2}}+\sin 2x+2x)+c\]                                

    D)            None of these

    Correct Answer: C

    Solution :

                       Put \[{{x}^{2}}+\sin 2x+2x=t,\] then it reduces to            \[\frac{1}{2}\int_{{}}^{{}}{\frac{1}{t}\,dt}=\frac{1}{2}\log t+c=\frac{1}{2}\log ({{x}^{2}}+\sin 2x+2x)+c.\]


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