Question:
If
\[
\int x \log\left(1 + \frac{1}{x}\right) \, dx = f(x) \log(x+1) + g(x) x^2 + Lx + C,
\]
then
If
\[ \int x \log\left(1 + \frac{1}{x}\right) \, dx = f(x) \log(x+1) + g(x) x^2 + Lx + C, \]
then
\[ \int x \log\left(1 + \frac{1}{x}\right) \, dx = f(x) \log(x+1) + g(x) x^2 + Lx + C, \]
then
Show Hint
Look at the coefficient of the logarithmic term after integration.
Updated On: Mar 23, 2026
- \(f(x)=\dfrac12x^2\)
- \(g(x)=\log x\)
- \(L=1\)
- None of these
Show Solution
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The Correct Option is A
Solution and Explanation
Integrating by parts,
\[ \int x \log\left(1 + \frac{1}{x}\right) \, dx = \frac{x^2}{2} \log\left(1 + \frac{1}{x}\right) + \cdots \]
Thus \(f(x) = \frac{1}{2} x^2\).
\[ \int x \log\left(1 + \frac{1}{x}\right) \, dx = \frac{x^2}{2} \log\left(1 + \frac{1}{x}\right) + \cdots \]
Thus \(f(x) = \frac{1}{2} x^2\).
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