A) \[\frac{x}{2}\sqrt{{{x}^{2}}+{{a}^{2}}}+\frac{{{a}^{2}}}{2}\log \{x+\sqrt{{{x}^{2}}+{{a}^{2}}}\}+c\]
B) \[\frac{x}{2}\sqrt{{{x}^{2}}+{{a}^{2}}}-\frac{{{a}^{2}}}{2}\log \{x+\sqrt{{{x}^{2}}+{{a}^{2}}}\}+c\]
C) \[\frac{x}{2}+\sqrt{{{x}^{2}}+{{a}^{2}}}+\frac{{{a}^{2}}}{2}\log \{x-\sqrt{{{x}^{2}}+{{a}^{2}}}\}+c\]
D) \[\sqrt{{{x}^{2}}+{{a}^{2}}}+\frac{{{a}^{2}}}{2}\log \{x-\sqrt{{{x}^{2}}+{{a}^{2}}}\}+c\]
Correct Answer: A
Solution :
\[\int{\sqrt{{{x}^{2}}+{{a}^{2}}}}dx\] \[=\frac{1}{2}x\sqrt{{{x}^{2}}+{{a}^{2}}}+\frac{{{a}^{2}}}{2}\log \{x+\sqrt{{{x}^{2}}+{{a}^{2}}}\}+c\]
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