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\int\frac{e^{5 \log_{e}x}- e^{4 \log_{e}x}}{e^{3 \log_{e}x}-e^{2 \log_{e}x}}.dx 

is equal to ?

  • Option 1)

    \frac{x^{3}}{3}+c

  • Option 2)

    \frac{x^{2}}{3}+c

  • Option 3)

    \frac{x^{4}}{3}+c

  • Option 4)

    \frac{x^{3}}{2}+c

 

Answers (1)

best_answer

 

Indefinite integrals for Algebraic functions -

 \frac{\mathrm{d}}{\mathrm{d} x} \frac{\left ( x^{n+1} \right )}{n+1}=x^{n} so \int x^{n}dx=\frac{x^{n+1}}{n+1}

- wherein

Where  n\neq-1

 

 \int \frac{e^{5\log_{e}x}-e^{4\log_{e}x}}{e^{3\log_{e}x}-e^{2\log_{e}x}}dx

=\int \frac{e^{log_{e}x^{5}}-e^{log_{e}x^{4}}}{e^{log_{e}x^{3}}-e^{log_{e}x^{2}}}dx 

\left [ \because e^{log_{e}x} =x\right ]

=\int \frac{x^{5}-x^{4}}{x^{3}-x^{2}}dx

=\int \frac{x^{4}(x-1)}{x^{2}(x-1)}dx

=\int x^{2}dx

=\frac{x^{3}}{3}+C

 


Option 1)

\frac{x^{3}}{3}+c

Option is Correct

Option 2)

\frac{x^{2}}{3}+c

Option is incorrect

Option 3)

\frac{x^{4}}{3}+c

Option is incorrect

Option 4)

\frac{x^{3}}{2}+c

Option is incorrect

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prateek

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