4 ∫ 1/√(3+x^2) + √(1+x^2) dx x ∈ to (0, 1) - 3In√3 is equal to

Question

\(4 \int_\limits{0}^{1} \frac{1}{\sqrt{3+x^{2}}+\sqrt{1+x^{2}}} d x-3 \ln \sqrt{3}\) is equal to

(1) \(3-\sqrt{2}+\ln (\sqrt{2}+1)\)

(2) \(2+\sqrt{2}-\ln (\sqrt{3}+1)\)

(3) \(2-\sqrt{2}-\ln (\sqrt{2}+1)\)

(4) \(2-\sqrt{3}-\ln (\sqrt{3}+1)\)

Answer 1

Correct option is: (3) \(2-\sqrt{2}-\ln (\sqrt{2}+1)\)    

\(I=4 \int_\limits{0}^{1} \frac{1}{\sqrt{3+x^{2}}+\sqrt{1+x^{2}}} d x\)

\(=2 \int_\limits{0}^{1} \sqrt{3+x^{2}}-\sqrt{1+x^{2}} d x\)

\(=2\left[\int_\limits{0}^{1} \sqrt{3+x^{2}} d x-\int_\limits{0}^{1} \sqrt{1+x^{2}} d x\right]\)

\(=2\left[\left(\frac{1}{2} x \sqrt{x^{2}+3}+\frac{3}{2} \ln \left|\sqrt{3+x^{2}}+x\right|\right)-\right. \left.\left(\frac{1}{2} x \sqrt{1+x^{2}}+\frac{1}{2} \ln \left|\sqrt{1+x^{2}}+x\right|\right)\right]_{0}^{1}\)

\(=2\left[\left(1+\frac{3}{2} \ln 3-\frac{3}{2} \ln \sqrt{3}\right)-\left(\frac{\sqrt{2}}{2}+\frac{1}{2} \ln (\sqrt{2}+1)\right)\right]\)

\(=2\left(1+\frac{3}{4} \ln 3-\frac{1}{\sqrt{2}}-\frac{1}{2} \ln (\sqrt{2}+1)\right)\)

\(=3 \ln \sqrt{3}+2-\sqrt{2}-\ln (\sqrt{2}+1)\)

\(I-3 \ln \sqrt{3}=2-\sqrt{2}-\ln (\sqrt{2}+1) \)