JEE Main 2021 (Online) 3rd August Morning ShiftQuestion 1
If $\int\frac{4x+13}{\sqrt{x^{2}+5x+6}}dx=A\sqrt{x^{2}+5x+6}+B\log_e\left|\left(x+\frac{5}{2}\right)+\sqrt{x^{2}+5x+6}\right|+C$ ($C$ is a constant of integration), then the ordered pair $(A,B)$ is equal to
Indefinite Integration
Solution
Concept. Write the numerator as a multiple of the derivative of $x^2+5x+6$ plus a constant. Formulas used. (1) $\int\dfrac{2x+5}{\sqrt{x^2+5x+6}}dx=2\sqrt{x^2+5x+6}$(2) $\int\dfrac{dx}{\sqrt{X^2-a^2}}=\ln\big|X+\sqrt{X^2-a^2}\big|$, with $x^2+5x+6=\big(x+\frac52\big)^2-\frac14$Step 1 - Split. $4x+13=2(2x+5)+3$. Step 2 - Integrate. By (1), (2), $$I=4\sqrt{x^2+5x+6}+3\ln\Big|x+\frac52+\sqrt{x^2+5x+6}\Big|+C.$$ So $(A,B)=(4,3)$. Answer: (D)