AP EAMCET · Maths · Indefinite Integration
\(\int \frac{d x}{(x-1) \sqrt{x+2}}=\)
- A \(\frac{2}{\sqrt{3}} \log \left|\frac{\sqrt{(x+2)}+\sqrt{3}}{\sqrt{(x+2)}-\sqrt{3}}\right|+c\)
- B \(\frac{-1}{\sqrt{3}} \log \left|\frac{\sqrt{(x+2)}-\sqrt{3}}{\sqrt{(x+2)}+\sqrt{3}}\right|+c\)
- C \(\frac{1}{\sqrt{3}} \log \left|\frac{\sqrt{(x+2)}+\sqrt{3}}{\sqrt{(x+2)}-\sqrt{3}}\right|+c\)
- D \(\frac{1}{\sqrt{3}} \log \left|\frac{\sqrt{(x+2)}-\sqrt{3}}{\sqrt{(x+2)}+\sqrt{3}}\right|+c\)
Answer & Solution
Correct Answer
(D) \(\frac{1}{\sqrt{3}} \log \left|\frac{\sqrt{(x+2)}-\sqrt{3}}{\sqrt{(x+2)}+\sqrt{3}}\right|+c\)
Step-by-step Solution
Detailed explanation
Let \(I=\int \frac{d x}{(x-1) \sqrt{x+2}}\) Let \(x=t^2-2\) or \(t=\sqrt{x+2} \Rightarrow \mathrm{d} x=2 t d t\) Hence \(I=\int \frac{2 t d t}{\left(\mathrm{t}^2-3\right) \mathrm{t}}=\frac{2}{2 \sqrt{3}} \log \left|\frac{t-\sqrt{3}}{t+\sqrt{3}}\right|+C\)…
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