AP EAMCET · Maths · Indefinite Integration
\(\int \frac{2 \tan (x)}{1+2 \tan ^2(x)} d x=\)
- A \(\log \left|\cos ^2 x+\sin ^2 x\right|+c\)
- B \(\log \left|\frac{\cos ^2 x}{2}+\sin ^2 x\right|+c\)
- C \(\log \left|\cos ^2 x+\frac{\sin ^2 x}{2}\right|+c\)
- D \(\log \left|\frac{\cos ^2 x}{2}+\frac{\sin ^2 x}{2}\right|+c\)
Answer & Solution
Correct Answer
(B) \(\log \left|\frac{\cos ^2 x}{2}+\sin ^2 x\right|+c\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \int\left(\frac{2 \tan x}{1+2 \tan ^2 x}\right) d x=\int \frac{2 \sin x \cdot \cos x}{\cos ^2 x+2 \sin ^2 x} d x \\ & \Rightarrow \quad \int \frac{2 \sin x \cdot \cos x}{1+\sin ^2 x} d x \end{aligned}\) Let…
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