AP EAMCET · Maths · Indefinite Integration
If \(f(x)=\int \frac{d x}{\left(x^2+2\right)}\) and \(f(\sqrt{2})=0\), then \(f(0)=\)
- A \(\frac{\pi}{2 \sqrt{2}}\)
- B \(\frac{-\pi}{2 \sqrt{2}}\)
- C \(\frac{-\pi}{4 \sqrt{2}}\)
- D \(\frac{\pi}{4 \sqrt{2}}\)
Answer & Solution
Correct Answer
(C) \(\frac{-\pi}{4 \sqrt{2}}\)
Step-by-step Solution
Detailed explanation
\(\quad f(x)=\int \frac{d x}{x^2+(\sqrt{2})^2}=\frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{x}{\sqrt{2}}\right)+c\) Since at \(x=\sqrt{2}, f(x)=0\) \[ \Rightarrow 0=\frac{1}{\sqrt{2}} \tan ^{-1}(1)+c \Rightarrow c=-\frac{\pi}{4 \sqrt{2}} \] Hence…
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