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\(\int \frac{d x}{\cos ^2(x)+\sin (2 x)}=\)
Options:
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1035 Upvotes
Verified Answer
The correct answer is:
\(\frac{1}{2} \log |1+2 \tan (x)|+c\)
\(I=\int \frac{d x}{\cos ^2 x+\sin 2 x}=\int \frac{\sec ^2 x d x}{1+2 \tan x}\)
Put \(1+2 \tan x=t \Rightarrow \sec ^2 x d x=\frac{d t}{2}\)
So, \(I=\frac{1}{2} \int \frac{d t}{t}=\frac{1}{2} \log _e|t|+C=\frac{1}{2} \log _e|1+2 \tan x|+C\)
Hence, option (c) is correct.
Put \(1+2 \tan x=t \Rightarrow \sec ^2 x d x=\frac{d t}{2}\)
So, \(I=\frac{1}{2} \int \frac{d t}{t}=\frac{1}{2} \log _e|t|+C=\frac{1}{2} \log _e|1+2 \tan x|+C\)
Hence, option (c) is correct.
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