The integral
Question:

The integral $\int \frac{e^{3 \log _{e} 2 x}+5 e^{2 \log _{e} 2 x}}{e^{4 \log _{e} x}+5 e^{3 \log _{e} x}-7 e^{2 \log _{e} x}} d x, x>0$, is equal to: (where $c$ is a constant of integration)

  1. (1) $\log _{e}\left|x^{2}+5 x-7\right|+c$

  2. (2) $\frac{1}{4} \log _{e}\left|x^{2}+5 x-7\right|+c$

  3. (3) $4 \log _{\mathrm{e}}\left|\mathrm{x}^{2}+5 \mathrm{x}-7\right|+\mathrm{c}$

  4. (4) $\log _{e} \sqrt{x^{2}+5 x-7}+c$


Correct Option: , 3

Solution:

$\int \frac{e^{3 \log _{e} 2 x}+5 e^{2 \log _{e} 2 x}}{e^{4 \log _{e} x}+5 e^{3 \log _{e} x}-7 e^{2 \log _{e} x}} d x$

$=\int \frac{8 x^{3}+5\left(4 x^{2}\right)}{x^{4}+5 x^{3}-7 x^{2}}$

$=\int \frac{8 x^{3}+20 x^{2}}{x^{4}+5 x^{3}-7 x^{2}}$

$=\int \frac{8 x+20}{x^{2}+5 x-7}$

$=\int \frac{4(2 x+5)}{x^{2}+5 x-7}$

$=\int \frac{4 d t}{t}$

$=4 \ln |t|+C$

$=4 \ln \left|\left(x^{2}+5 x-7\right)\right|+c$

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