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

  • question_answer
    The value of\[\int_{{}}^{{}}{{{e}^{x}}{{\sec }^{2}}({{e}^{x}})\ dx}\] is

    A)            \[\tan ({{e}^{x}})+k\]

    B)            \[\tan ({{e}^{x}})\ .\ e+k\]

    C)            \[{{e}^{x}}\tan x+k\]

    D)            \[\frac{\tan ({{e}^{x}})}{{{e}^{x}}}+k\]

    Correct Answer: A

    Solution :

               \[I=\int_{{}}^{{}}{{{e}^{x}}{{\sec }^{2}}({{e}^{x}})\,dx}\]. Put \[{{e}^{x}}=t\Rightarrow {{e}^{x}}dx=dt\]            \[\therefore \,\,\,I=\int_{{}}^{{}}{{{\sec }^{2}}t\,dt=\tan t+k=\tan ({{e}^{x}})+k}\].


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