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JEE Main & Advanced Mathematics Indefinite Integrals Question Bank Critical Thinking

  • question_answer
    \[\int_{{}}^{{}}{\sin \sqrt{x}}\ dx=\]       [Roorkee 1977]

    A) \[2[\sin \sqrt{x}-\cos \sqrt{x}]+c\]       

    B) \[2[\sin \sqrt{x}-\sqrt{x}\cos \sqrt{x}]+c\]

    C) \[2[\sin \sqrt{x}+\cos \sqrt{x}]+c\]       

    D) \[2[\sin \sqrt{x}+\sqrt{x}\cos \sqrt{x}]+c\]

    Correct Answer: B

    Solution :

    • Put \[\sqrt{x}=t\Rightarrow \frac{1}{2\sqrt{x}}\,dx=dt\Rightarrow dx=2t\,dt,\] then                   
    • \[\int_{{}}^{{}}{\sin \sqrt{x}\,dx}=2\int_{{}}^{{}}{t\sin t\,dt}=2(-t\cos t+\sin t)+c\]                                       
    • \[=2(\sin \sqrt{x}-\sqrt{x}\cos \sqrt{x})+c.\]


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