4 _ 0 ^ 1 (1 3+x^ 2 + 1+x^ 2 ) dx-3 _ c (3) is equal to : - Vidyadip

MCQ

$4 \int_{0}^{1}\left(\frac{1}{\sqrt{3+\mathrm{x}^{2}}+\sqrt{1+\mathrm{x}^{2}}}\right) \mathrm{dx}-3 \log _{\mathrm{c}}(\sqrt{3})$ is equal to :

A
$2+\sqrt{2}+\log _{\mathrm{e}}(1+\sqrt{2})$
B
$2-\sqrt{2}-\log _{e}(1+\sqrt{2})$
C
$2+\sqrt{2}-\log _{\mathrm{e}}(1+\sqrt{2})$
D
$2-\sqrt{2}+\log _{\mathrm{e}}(1+\sqrt{2})$

✓

Answer

B. $2-\sqrt{2}-\log _{c}(1+\sqrt{2})$
$4 \int_{0}^{1} \frac{1}{\sqrt{3+\mathrm{x}^{2}}+\sqrt{1+\mathrm{x}^{2}}} \mathrm{dx}-3 \ln \sqrt{3}$
$=4 \int_{0}^{1} \frac{\sqrt{3+\mathrm{x}^{2}}-\sqrt{1+\mathrm{x}^{2}}}{\left(3+\mathrm{x}^{2}\right)-\left(1-\mathrm{x}^{2}\right)} \mathrm{dx}-\frac{3}{2} \ln 3$
$=2\left[\left\{\frac{\mathrm{x}}{2} \sqrt{3+\mathrm{x}^{2}}+\frac{3}{2} \ln \left(\mathrm{x}+\sqrt{3+\mathrm{x}^{2}}\right)\right\}_{0}^{1}\right.$
$\left.-\left\{\frac{\mathrm{x}}{2} \sqrt{1+\mathrm{x}^{2}}+\frac{1}{2} \ln \left(\mathrm{x}+\sqrt{1+\mathrm{x}^{2}}\right)\right\}_{0}^{1}\right]-\frac{3}{2} \ln 3$
$=2\left[\left\{\frac{1}{2} \sqrt{4}+\frac{3}{2} \ln (1+\sqrt{4})\right\}-\left\{0+\frac{3}{2} \ln \sqrt{3}\right\}\right.$
$\left.-\left\{\frac{1}{2} \sqrt{2}+\frac{1}{2} \ln (1+\sqrt{2})\right\}+\left\{0+\frac{1}{2}(0)\right\}\right]-\frac{3}{2} \ln 3$
$=2\left[1+\frac{3}{2} \ln 3-\frac{3}{4} \ln 3-\frac{1}{\sqrt{2}}-\frac{1}{2} \ln (1+\sqrt{2})\right]-\frac{3}{2} \ln 3$
$=2+3 \ln 3-\frac{3}{2} \ln 3-\sqrt{2}-\ln (1+\sqrt{2})-\frac{3}{2} \ln 3$
$=2-\sqrt{2}-\ln (1+\sqrt{2})$

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