A) \[\frac{{{(1+{{x}^{2}})}^{3/2}}}{3}\sqrt{1+{{x}^{2}}}+c\]
B) \[{{x}^{2}}\sqrt{1+{{x}^{2}}}-\frac{1}{3}\sqrt{{{(1+{{x}^{2}})}^{3}}}+c\]
C) \[\frac{1}{3}{{x}^{2}}\sqrt{1+{{x}^{2}}}-\frac{2}{3}\sqrt{{{(1+{{x}^{2}})}^{3}}}+c\]
D) none of these
Correct Answer: C
Solution :
| \[1+{{x}^{2}}={{t}^{2}}\] |
| \[\Rightarrow \]\[l=\int{({{t}^{2}}-1)\,dt=\frac{{{t}^{3}}}{3}-t=\frac{{{(1+{{x}^{2}})}^{3/2}}}{3}}-\sqrt{1+{{x}^{2}}}\]\[=\frac{(1+{{x}^{2}})\sqrt{1+{{x}^{2}}}}{3}-\sqrt{1+{{x}^{2}}}\,(1+{{x}^{2}}-{{x}^{2}})\]\[=\frac{1}{3}{{x}^{2}}\sqrt{1+{{x}^{2}}}-\frac{2}{3}{{(1+{{x}^{2}})}^{3/2}}\] |
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