Q36 / 90 · JEE Main 2021 (25 Feb Shift 2) · Mathematics

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)

Solution

$\int \frac{e^{3 \log_{e} 2 x} + 5 e^{2 \log_{e} 2 x}}{e^{4 \log_{e} x} + 5 e^{3 \log_{e} x} - 7 e^{2 \log_{e} x}} d x, x > 0$

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

$= 4 \int \frac{d (x^{2} + 5 x - 7)}{(x^{2} + 5 x - 7)} = 4 \log_{e} |x^{2} + 5 x - 7| + c$

Hence, the correct answer is option: 'B'.