TS EAMCET · Maths · Indefinite Integration
\(\int \frac{\log x}{(1+x)^3} d x=\)
- A \(\frac{1}{2}\left[\frac{1}{1+x}+\frac{\log x}{(1+x)^2}-\log \left(x^2+x\right)\right]+\mathrm{c}\)
- B \(\frac{1}{2}\left[\frac{1}{1+x}-\frac{\log x}{(1+x)}-\log \left(1+x^2\right)\right]+\mathrm{c}\)
- C \(\frac{1}{2}\left[\frac{1}{1+x}+\frac{\log x}{(1+x)^2}-\log \left(1+x^2\right)\right]+\mathrm{c}\)
- D \(\frac{1}{2}\left[\frac{1}{1+x}-\frac{\log x}{(1+x)^2}+\log \left(\frac{x}{1+x}\right)\right]+\mathrm{c}\)
Answer & Solution
Correct Answer
(D) \(\frac{1}{2}\left[\frac{1}{1+x}-\frac{\log x}{(1+x)^2}+\log \left(\frac{x}{1+x}\right)\right]+\mathrm{c}\)
Step-by-step Solution
Detailed explanation
Step 1: Use integration by parts \(\int u \, dv = uv - \int v \, du\) with \(u = \log x\) and \(dv = (1+x)^{-3} dx\). \(du = \frac{1}{x} dx\), \(v = -\frac{1}{2(1+x)^2}\).…
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