JEE PYQ: Indefinite Integration Question 6
Question 6 - 2021 (25 Feb Shift 2)
The integral $\int \frac{e^{3\log_e 2x} + 5e^{2\log_e 2x}}{e^{4\log_e x} + 5e^{3\log_e x} - 7e^{2\log_e x}},dx$, $x > 0$, is equal to (where $c$ is a constant of integration):
(1) $\log_e|x^2 + 5x - 7| + c$
(2) $\frac{1}{4}\log_e|x^2 + 5x - 7| + c$
(3) $4\log_e|x^2 + 5x - 7| + c$
(4) $\log_e\sqrt{x^2 + 5x - 7} + c$
Show Answer
Answer: (3)
Solution
Simplify: numerator $= 8x^3 + 20x^2 = 4x^2(2x+5)$, denominator $= x^4 + 5x^3 - 7x^2 = x^2(x^2+5x-7)$. So integral $= \int \frac{4(2x+5)}{x^2+5x-7},dx = 4\ln|x^2+5x-7| + c$.
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