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GATE 2025 CE (CE2) – Question 12

Engineering Mathematics · Calculus: Integrals, partial and total derivatives, gradient, divergence, curl, line, surface and volume integrals · 1 mark · Multiple choice

Integration of $\ln(x)$ with $x$ i.e., $\int \ln(x)\,dx = $ ______.

  1. $x\ln(x) - x + Constant$
  2. $x - \ln(x) + Constant$
  3. $x\ln(x) + x + Constant$
  4. $\ln(x) - x + Constant$

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Correct answer: (A) $x\ln(x) - x + Constant$

Explanation

Integrate by parts with $u = \ln x$ and $dv = dx$: $\int \ln x\,dx = x\ln x - \int x \cdot \frac{1}{x}dx = x\ln x - x + C$.