TS EAMCET · Maths · Indefinite Integration
If \(\int(\sqrt{\operatorname{cosec} x+1}) d x=k \tan ^{-1}(f(x))+c\), then \(\frac{1}{k} f\left(\frac{\pi}{6}\right)=\)
- A \(\frac{1}{2}\)
- B \(\frac{1}{4}\)
- C \(-\frac{1}{4}\)
- D \(-\frac{1}{2}\)
Answer & Solution
Correct Answer
(D) \(-\frac{1}{2}\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \mathrm{I}=\int \sqrt{\operatorname{cosec} x+1} d x \\ = & \int \sqrt{\frac{\tan ^2\left(\frac{x}{2}\right)+1}{2 \tan \left(\frac{x}{2}\right)}+1} d x\end{aligned}\) \(\operatorname{Let} u=\tan \frac{x}{2} \Rightarrow d x=\frac{2}{u^2+1} d u\)…
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