JEE PYQ: Indefinite Integration Question 25
Question 25 - 2019 (12 Apr Shift 1)
The integral $\int \frac{2x^3 - 1}{x^4 + x},dx$ is equal to (Here $C$ is a constant of integration):
(1) $\frac{1}{2}\log_e\left|\frac{x^3+1}{x^2}\right| + C$
(2) $\frac{1}{2}\log_e\frac{(x^3+1)^2}{|x^3|} + C$
(3) $\log_e\left|\frac{x^3+1}{x}\right| + C$
(4) $\log_e\left|\frac{x^3+1}{x^2}\right| + C$
Show Answer
Answer: (3)
Solution
$\frac{2x^3-1}{x^4+x} = \frac{2x^3-1}{x(x^3+1)} = \frac{3x^2}{x^3+1} - \frac{1}{x}$ (partial fractions: $\frac{2x^3-1}{x(x^3+1)} = \frac{A}{x} + \frac{Bx^2+Cx+D}{x^3+1}$, giving $-\frac{1}{x} + \frac{3x^2}{x^3+1} - …$). Integrating: $\ln|x^3+1| - \ln|x| + C = \log_e\left|\frac{x^3+1}{x}\right| + C$.
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