COMEDK · Maths · 28. Indefinite Integration
\(\int \dfrac{\log x}{(1 + x)^2}\, dx\)
- A \(\dfrac{\log x}{x + 1} - \log\left|\dfrac{x}{x + 1}\right| + C\)
- B \(-\dfrac{\log x}{x + 1} - \log\left|\dfrac{x}{x + 1}\right| + C\)
- C \(-\dfrac{\log x}{x + 1} + \log\left|\dfrac{x}{x + 1}\right| + C\)
- D \(\dfrac{\log x}{x + 1} + \log\left|\dfrac{x + 1}{x}\right| + C\)
Answer & Solution
Correct Answer
(C) \(-\dfrac{\log x}{x + 1} + \log\left|\dfrac{x}{x + 1}\right| + C\)
Step-by-step Solution
Detailed explanation
Using integration by parts: \(\int \dfrac{\log x}{(1 + x)^2} dx\) Let \(u = \log x\) and \(dv = \dfrac{1}{(1 + x)^2} dx\) \(du = \dfrac{1}{x} dx\) and \(v = -\dfrac{1}{1 + x}\) \(I = -\dfrac{\log x}{1 + x} - \int \left(-\dfrac{1}{1 + x}\right) \dfrac{1}{x} dx\)…
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