### Related Formula int x^n dx = fracx^n+1n+1$\int x^n dx = \frac{x^{n+1}}{n+1}$ ### Core Logic To evaluate integrals with sum of square roots in the denominator, multiply and divide by the conjugate to rationalize it. I = int_0^1fracsqrt3+x-sqrt1+x(sqrt3+x+sqrt1+x)(sqrt3+x-sqrt1+x)dx$I = \int_{0}^{1}\frac{\sqrt{3+x}-\sqrt{1+x}}{(\sqrt{3+x}+\sqrt{1+x})(\sqrt{3+x}-\sqrt{1+x})}dx$ I = int_0^1fracsqrt3+x-sqrt1+x(3+x) - (1+x)dx$I = \int_{0}^{1}\frac{\sqrt{3+x}-\sqrt{1+x}}{(3+x) - (1+x)}dx$ I = frac12 int_0^1 (sqrt3+x - sqrt1+x) dx$I = \frac{1}{2} \int_{0}^{1} (\sqrt{3+x} - \sqrt{1+x}) dx$ ### Step 1: Integration and Bounds Setup Integrate the resulting expression: I = frac12 left[ frac(3+x)^3/23/2 - frac(1+x)^3/23/2 right]_0^1$I = \frac{1}{2} \left[ \frac{(3+x)^{3/2}}{3/2} - \frac{(1+x)^{3/2}}{3/2} \right]_{0}^{1}$ I = frac12 cdot frac23 left[ (3+x)^3/2 - (1+x)^3/2 right]_0^1$I = \frac{1}{2} \cdot \frac{2}{3} \left[ (3+x)^{3/2} - (1+x)^{3/2} \right]_{0}^{1}$ I = frac13 left[ ((4)^3/2 - (2)^3/2) - ((3)^3/2 - (1)^3/2) right]$I = \frac{1}{3} \left[ ((4)^{3/2} - (2)^{3/2}) - ((3)^{3/2} - (1)^{3/2}) \right]$ ### Step 2: Term Simplification Evaluate the boundary powers: 4^3/2 = 8$4^{3/2} = 8$ 2^3/2 = 2sqrt2$2^{3/2} = 2\sqrt{2}$ 3^3/2 = 3sqrt3$3^{3/2} = 3\sqrt{3}$ 1^3/2 = 1$1^{3/2} = 1$ Substitute back into the expression: I = frac13 [ 8 - 2sqrt2 - 3sqrt3 + 1 ] = frac13 [ 9 - 2sqrt2 - 3sqrt3 ]$I = \frac{1}{3} [ 8 - 2\sqrt{2} - 3\sqrt{3} + 1 ] = \frac{1}{3} [ 9 - 2\sqrt{2} - 3\sqrt{3} ]$ I = 3 - frac23sqrt2 - sqrt3$I = 3 - \frac{2}{3}\sqrt{2} - \sqrt{3}$ ### Step 3: Finding Co-efficients Comparing with a+bsqrt2+csqrt3$a+b\sqrt{2}+c\sqrt{3}$ yields: a = 3$a = 3$, b = -frac23$b = -\frac{2}{3}$, c = -1$c = -1$ Compute 2a+3b-4c$2a+3b-4c$: 2(3) + 3left(-frac23right) - 4(-1)$2(3) + 3\left(-\frac{2}{3}\right) - 4(-1)$ 6 - 2 + 4 = 8$6 - 2 + 4 = 8$ ### Pattern Recognition Whenever you see a sum of square roots in the denominator of an integrand, the immediate algorithmic next step is rationalization. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Maths: Definite Integration