QJEEM24Mad5f8: JEE Main 2024 Mathematics - Class 12 JEEM24Mad5f8 Solution: If , where are rational numbers, then is equal to :

$$\begin{array}{rl}& \underset{0}{\overset{1}{\int }}\frac{1}{\sqrt{3+x}+\sqrt{1+x}}dx=\underset{0}{\overset{1}{\int }}\frac{\sqrt{3+x}-\sqrt{1+x}}{(3+x)-(1+x)}dx\\ & \frac{1}{2}[\underset{0}{\overset{1}{\int }}\sqrt{3+x}dx-\underset{0}{\overset{1}{\int }}(\sqrt{1+x})dx]\end{array}$$

$\begin{aligned}& \frac{1}{2}\left[2 \frac{(3+x)^{\frac{3}{2}}}{3}-\frac{2(1+x)^{\frac{3}{2}}}{3}\right]_0^1 \\& \frac{1}{2}\left[\frac{2}{3}(8-3 \sqrt{3})-\frac{2}{3}\left(2^{\frac{3}{2}}-1\right)\right] \\& \frac{1}{3}[8-3 \sqrt{3}-2 \sqrt{2}+1] \\& =3-\sqrt{3}-\frac{2}{3} \sqrt{2}=a+b \sqrt{2}+c \sqrt{3} \\& a=3, b=-\frac{2}{3}, c=-1 \\& 2 a+3 b-4 c=6-2+4=8\end{aligned}$