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JEE Main & Advanced Mathematics Definite Integration Question Bank Fundamental definite integration, Definite integration by substitution

  • question_answer
    \[\int_{1}^{2}{\frac{1}{{{x}^{2}}}{{e}^{\frac{-1}{x}}}\,dx=}\]                                 [DCE 2001]

    A)                 \[\sqrt{e}+1\]

    B)                 \[\sqrt{e}-1\]

    C)                 \[\frac{\sqrt{e}+1}{e}\]

    D)                 \[\frac{\sqrt{e}-1}{e}\]

    Correct Answer: D

    Solution :

                    Put \[t=-\frac{1}{x}\Rightarrow dt=\frac{1}{{{x}^{2}}}dx\], then it reduces to \[\int_{-1}^{-1/2}{{{e}^{t}}dt=[{{e}^{t}}]_{-1}^{-1/2}={{e}^{-1/2}}-{{e}^{-1}}}=\frac{\sqrt{e}-1}{e}\].


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