A) \[{{(\sqrt{2}-1)}^{2}}\]
B) \[\frac{{{(\sqrt{2}-1)}^{2}}}{\sqrt{2}}\]
C) \[\frac{\sqrt{2}-1}{\sqrt{2}}\]
D) None of these
Correct Answer: D
Solution :
Put \[t={{x}^{2}}+1\Rightarrow dt=2x\,dx\] \[\int_{0}^{2}{\frac{{{x}^{3}}}{{{({{x}^{2}}+1)}^{3/2}}}dx=\frac{1}{2}}\int_{1}^{5}{\frac{(t-1)}{{{t}^{3/2}}}dt=\frac{1}{2}\int_{1}^{5}{[{{t}^{-1/2}}-{{t}^{-3/2}}]\,dt}}\] \[=\frac{1}{2}\left[ 2\sqrt{t}+2\frac{1}{\sqrt{t}} \right]_{1}^{5}=\frac{1}{2}\left[ 2\sqrt{5}+\frac{2}{\sqrt{5}}-2-2 \right]\] \[=\left[ \sqrt{5}+\frac{1}{\sqrt{5}}-2 \right]=\frac{6-2\sqrt{5}}{\sqrt{5}}\].
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