$\underset{0}{\overset{1}{\int <!--\; \int \; -->}}\frac{1}{\sqrt{3+x}+\sqrt{1+x}}dx=a+b\sqrt{2}+c\sqrt{3},$ then 2a – 3b – 4c is equal to

The correct answer is 12.

Step-by-step Explanation:

Let the given definite integral be denoted by I:
$$I={\int }_0^1\frac{1}{\sqrt{x+3}+\sqrt{x+1}}dx$$

To evaluate this integral, we first rationalize the denominator by multiplying the numerator and denominator by the conjugate expression, which is:
$$\sqrt{x+3}-\sqrt{x+1}$$

Applying this rationalization, we get:
$$I={\int }_0^1\frac{\sqrt{x+3}-\sqrt{x+1}}{(\sqrt{x+3}+\sqrt{x+1})(\sqrt{x+3}-\sqrt{x+1})}dx$$
Using the algebraic identity $(a+b)(a-b)=a^2-b^2$ in the denominator:
$$I={\int }_0^1\frac{\sqrt{x+3}-\sqrt{x+1}}{(x+3)-(x+1)}dx$$
Simplifying the denominator:
$$I={\int }_0^1\frac{\sqrt{x+3}-\sqrt{x+1}}{2}dx$$

Now, split the integral into two separate parts:
$$I=\frac{1}{2}{\int }_0^1{(x+3)}^{1/2}dx-\frac{1}{2}{\int }_0^1{(x+1)}^{1/2}dx$$

Integrating each term using the power rule $\int u^{n}du=\frac{u^{n+1}}{n+1}$:
$$I=\frac{1}{2}{\left[\frac{{(x+3)}^{3/2}}{3/2}\right]}_0^1-\frac{1}{2}{\left[\frac{{(x+1)}^{3/2}}{3/2}\right]}_0^1$$
Simplifying the fractions:
$$I={\left[\frac{{(x+3)}^{3/2}}{3}\right]}_0^1-{\left[\frac{{(x+1)}^{3/2}}{3}\right]}_0^1$$

Substitute the upper limit (1) and lower limit (0) into each term:
$$I=\frac{1}{3}[4^{3/2}-3^{3/2}]-\frac{1}{3}[2^{3/2}-1^{3/2}]$$
Evaluating the powers:
$$4^{3/2}={\left(2^2\right)}^{3/2}=2^3=8$$
$$3^{3/2}=3\sqrt{3}$$
$$2^{3/2}=2\sqrt{2}$$
$$1^{3/2}=1$$
Substitute these simplified terms back into the expression:
$$I=\frac{1}{3}[8-3\sqrt{3}]-\frac{1}{3}[2\sqrt{2}-1]$$
Combine the terms under a single denominator:
$$I=\frac{8-3\sqrt{3}-2\sqrt{2}+1}{3}$$
$$I=\frac{9-2\sqrt{2}-3\sqrt{3}}{3}$$
$$I=3-\frac{2}{3}\sqrt{2}-\sqrt{3}$$

We are given that:
$$I=a+b\sqrt{2}+c\sqrt{3}$$
Comparing the coefficients, we obtain:
$$a=3$$
$$b=-\frac{2}{3}$$
$$c=-1$$

Now, find the values of 2a, 3b, and 4c:
$$2a=2\left(3\right)=6$$
$$3b=3(-\frac{2}{3})=-2$$
$$4c=4(-1)=-4$$

Finally, we compute the requested expression:
$$2a-3b-4c=6-(-2)-(-4)$$
$$2a-3b-4c=6+2+4=12$$
Therefore, the value of the expression is 12.