A) \[\frac{1}{2\sqrt{5}}\log \left[ \frac{\log x+2-\sqrt{5}}{\log x+2+\sqrt{5}} \right]+c\]
B) \[\frac{1}{\sqrt{5}}\log \left[ \frac{\log x+2-\sqrt{5}}{\log x+2+\sqrt{5}} \right]+c\]
C) \[\frac{1}{2\sqrt{5}}\log \left[ \frac{\log x+2+\sqrt{5}}{\log x+2-\sqrt{5}} \right]+c\]
D) \[\frac{1}{\sqrt{5}}\log \left[ \frac{\log x+2+\sqrt{5}}{\log x+2-\sqrt{5}} \right]+c\]
Correct Answer: A
Solution :
Put \[\log x=t\Rightarrow \frac{1}{x}\,dx=dt,\] then \[\int_{{}}^{{}}{\frac{dx}{x[{{(\log x)}^{2}}+4\log x-1]}}=\int_{{}}^{{}}{\frac{dt}{{{t}^{2}}+4t-1}}\] \[=\int_{{}}^{{}}{\frac{dt}{{{(t+2)}^{2}}-{{(\sqrt{5})}^{2}}}=\frac{1}{2\sqrt{5}}\log \left[ \frac{t+2-\sqrt{5}}{t+2+\sqrt{5}} \right]}\] \[=\frac{1}{2\sqrt{5}}\log \left[ \frac{\log x+2-\sqrt{5}}{\log x+2+\sqrt{5}} \right]+c\].
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