AP EAMCET · Maths · Indefinite Integration
\(\int \frac{d x}{\cos ^2(x)+\sin (2 x)}=\)
- A \(\frac{1}{2} \log |1+2 \cos (x)|+c\)
- B \(\frac{1}{2} \log |1-2 \tan (x)|+c\)
- C \(\frac{1}{2} \log |1+2 \tan (x)|+c\)
- D \(\frac{1}{2} \log |1+2 \cot (x)|+c\)
Answer & Solution
Correct Answer
(C) \(\frac{1}{2} \log |1+2 \tan (x)|+c\)
Step-by-step Solution
Detailed explanation
\(I=\int \frac{d x}{\cos ^2 x+\sin 2 x}=\int \frac{\sec ^2 x d x}{1+2 \tan x}\) 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.
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