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JAMIA MILLIA ISLAMIA Jamia Millia Islamia Solved Paper-2010

  • question_answer
        The value of\[\int{\frac{{{e}^{x}}\,dx}{\sqrt{5-4{{e}^{x}}-{{e}^{2x}}}}}\]is equal to

    A)  \[{{\tan }^{-1}}\left( \frac{{{e}^{x}}+2}{3} \right)+c\]

    B)  \[{{\sin }^{-1}}\left( \frac{{{e}^{x}}+2}{3} \right)+c\]

    C)  \[{{\cos }^{-1}}\left( \frac{{{e}^{x}}+2}{3} \right)+c\]

    D)  None of the above

    Correct Answer: B

    Solution :

                    Let \[I=\int{\frac{{{e}^{x}}\,dx}{\sqrt{5-4{{e}^{x}}-{{e}^{2x}}}}}\] Put\[{{e}^{x}}=t\] \[\Rightarrow \]                               \[{{e}^{x}}dx=dt\] \[\therefore \]                  \[I=\int{\frac{dt}{\sqrt{5-4t-{{t}^{2}}}}}\]                                 \[=\int{\frac{dt}{\sqrt{5({{t}^{2}}+4t+4)+4}}}\]                                 \[I=\int{\frac{dt}{\sqrt{9-{{(t+2)}^{2}}}}}\] \[\Rightarrow \]               \[I={{\sin }^{-1}}\left( \frac{t+2}{3} \right)+c\] \[\Rightarrow \]               \[I={{\sin }^{-1}}\left( \frac{{{e}^{x}}+2}{3} \right)+c\]


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