Indefinite Integration: The integral int(2x^3-1)/(x^4+x)dx is equal to | JEE Main 2019

JEE Main 2019 (Online) 12th April Morning ShiftQuestion 1

The integral $\int\frac{2x^3-1}{x^4+x}dx$ is equal to

Indefinite Integration

$\frac12\log_e\frac{\left(x^3+1\right)^2}{|x^3|}+C$$\log_e\frac{\left|x^3+1\right|}{x^2}+C$$\log_e\frac{x^3+1}{x}+C$✓ Correct answer$\frac12\log_e\frac{\left|x^3+1\right|}{x^2}+C$

Solution

Concept. After dividing by $x^2$, the numerator is the derivative of the denominator. Formulas used. (1) $\int\dfrac{h'(x)}{h(x)}\,dx=\ln|h(x)|+C$Step 1 - Divide by $x^2$. $$\frac{2x^3-1}{x^4+x}=\frac{2x-x^{-2}}{x^2+x^{-1}}.$$ Here $\dfrac{d}{dx}\left(x^2+\dfrac1x\right)=2x-\dfrac1{x^2}$. Step 2 - Use (1). $$\int=\ln\left|x^2+\frac1x\right|+C=\ln\left|\frac{x^3+1}{x}\right|+C.$$ Answer: (C)

Indefinite Integration - JEE Mathematics question (JEE Main 2019 (Online) 12th April Morning Shift)

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