A) \[\text{lo}{{\text{g}}_{e}}\left| \text{sec}\left( \frac{{{x}^{2}}-1}{2} \right) \right|+\text{c}\]
B) \[\text{lo}{{\text{g}}_{e}}\left| \frac{1}{2}\text{se}{{\text{c}}^{2}}({{x}^{2}}-1) \right|+\text{c}\]
C) \[\frac{1}{2}\text{lo}{{\text{g}}_{e}}\left| \text{se}{{\text{c}}^{2}}\left( \frac{{{x}^{2}}-1}{2} \right) \right|+\text{c}\]
D) \[\frac{1}{2}{{\log }_{e}}\left| \sec ({{x}^{2}}-1) \right|+c\]
Correct Answer: A
Solution :
| Given, \[{{x}^{2}}\,\ne \,n\pi +1,\]\[n\in N\] and we have, \[\left( \frac{2\,\text{sin}\theta -\text{sin}\,2\theta }{2\text{sin}\,\theta +\,\text{sin}2\theta \,} \right)\,=\,\left( \frac{1-\,\text{cos}\theta }{1+\text{cos}\theta } \right)\,=\,\text{ta}{{\text{n}}^{2}}\left( \frac{\theta }{2} \right)\] |
| \[\therefore \]\[\int{x\sqrt{\frac{2\,\text{sin}\,({{x}^{2}}-1)-\,\text{sin}2\,({{x}^{2}}-1)}{2\,\text{sin}\,({{x}^{2}}-1)\,-\,\text{sin}\,2\,({{x}^{2}}-1)}\,}}dx\] |
| \[=\,\int{x\text{tan}\,\left( \frac{{{x}^{2}}-1}{2} \right)dx}\] |
| \[=\,\int{\text{tan}\,\left( \frac{{{x}^{2}}-1}{2} \right)\,.d\,\left( \frac{{{x}^{2}}-1}{2} \right)}\] |
| \[=\,\text{log}\,\left| \text{sec}\,\left( \frac{{{x}^{2}}-1}{2} \right) \right|+c.\] |
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