AP EAMCET · Maths · Indefinite Integration
If \(\int \sqrt{\frac{2}{1+\sin x}} d x=2 \log |A(x)-B(x)|+C\) and \(0 \leq x \leq \frac{\pi}{2}\) then \(B\left(\frac{\pi}{4}\right)=\)
- A \(\frac{1}{\sqrt{2+3 \sqrt{3}}}\)
- B \(\frac{1}{\sqrt{3+2 \sqrt{2}}}\)
- C \(\frac{-1}{\sqrt{3+2 \sqrt{2}}}\)
- D \(\frac{2}{\sqrt{2+\sqrt{2}}}\)
Answer & Solution
Correct Answer
(B) \(\frac{1}{\sqrt{3+2 \sqrt{2}}}\)
Step-by-step Solution
Detailed explanation
\(I=\int \sqrt{\frac{2}{1+\sin x}} d x \Rightarrow I=\int \sqrt{\frac{2}{1+\cos \left(x-\frac{\pi}{2}\right)}} d x\) \(I=\int \frac{1}{\cos \left(\frac{x}{2}-\frac{\pi}{4}\right)} d x \Rightarrow I=\int \sec \left(\frac{x}{2}-\frac{\pi}{4}\right) d x\)…
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