TS EAMCET · Maths · Indefinite Integration
\(\int \frac{\sec x}{3(\sec x+\tan x)+2} d x=\)
- A \(\frac{1}{2} \log \left|\frac{\tan \frac{x}{2}+1}{\tan \frac{x}{2}+5}\right|+c\)
- B \(\frac{2}{\sqrt{11}} \tan ^{-1}\left(\frac{3 \tan \frac{x}{2}+4}{\sqrt{11}}\right)+c\)
- C \(\log |3 \sec x+2 \tan x|+c\)
- D \(\log |3 \tan x+2 \sec x|+c\)
Answer & Solution
Correct Answer
(A) \(\frac{1}{2} \log \left|\frac{\tan \frac{x}{2}+1}{\tan \frac{x}{2}+5}\right|+c\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & I=\int \frac{\sec x(\sec x-\tan x) d x}{3\left(\sec ^2 x-\tan ^2 x\right)+2(\sec x-\tan x)} \\ & \sec x-\tan x=t \\ & I=\int \frac{-d t}{3+2 t}=\frac{-1}{2} \ln (3+2 t)+C \\ &= \frac{-1}{2} \ln (3+2 \sec x-2 \tan x)+C \\ &= \frac{-1}{2} \ln \left(\frac{3 \cos…
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