Integral Calculus Question 163
Question: If $ \int{\sec x\cos ecxdx=\log | g(x) |}+c, $ then what is $ g(x) $ equal to?
Options:
A) $ \sin x\cos x $
B) $ {{\sec }^{2}}x $
C) $ \tan x $
D) $ \log | \tan x | $
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Answer:
Correct Answer: C
Solution:
[c] Let $ I=\int{\sec x.\cos ecxdx} $ $ =\int{\frac{2}{2\sin x\cos x}dx} $ $ =2\int{\frac{2}{\sin 2x}dx-2\int{\frac{1}{\frac{2\tan x}{1+{{\tan }^{2}}x}}}} $ $ [\because \sin 2x=\frac{2\tan x}{1+{{\tan }^{2}}x}] $ $ =\int{\frac{{{\sec }^{2}}x}{\tan x}dx} $ Let $ \tan x=t\Rightarrow {{\sec }^{2}}dx=dt $ So, $ I=\int{\frac{dt}{t}=\log | t |+c=\log | \tan x |+c} $ But $ \int{\sec x\cos ecxdx=\log | g(x) |+c} $
$ \therefore g(x)=\tan x $
BETA
