A) \[\log \left[ \tan x+\sqrt{{{\tan }^{2}}x+4} \right]+c\]
B) \[\frac{1}{2}\log \left[ \tan x+\sqrt{{{\tan }^{2}}x+4} \right]+c\]
C) \[\log \left[ \frac{1}{2}\tan x+\frac{1}{2}\sqrt{{{\tan }^{2}}x+4} \right]+c\]
D) None of these
Correct Answer: A
Solution :
Put \[t=\tan x\Rightarrow dt={{\sec }^{2}}x\,dx,\] then \[\int_{{}}^{{}}{\frac{{{\sec }^{2}}x\,dx}{\sqrt{{{\tan }^{2}}x+4}}}=\int_{{}}^{{}}{\frac{1}{\sqrt{{{t}^{2}}+{{2}^{2}}}}}\,dt\] \[=\log [\tan x+\sqrt{{{\tan }^{2}}x+4}]+c.\]
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