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CA Foundation · Quantitative Aptitude · Differential and Integral Calculus

Evaluate ∫ x² log x dx.

The integral is (x³/3) log x − x³/9 + C. Using parts with u = log x and dv = x² dx, the remaining integral is (1/3)∫x² dx = x³/9, which is subtracted from the first term.

  1. A(x³/3) log x − x³/9 + CCorrect
  2. B(x³/3) log x − x³/3 + C
  3. C(x³/3) log x + x³/9 + C
  4. Dx³ log x/3 − x²/9 + C

Explanation

Take u = log x and dv = x² dx, so v = x³/3. Then the integral is (x³/3) log x − ∫(x³/3)(1/x) dx = (x³/3) log x − (1/3)(x³/3) = (x³/3) log x − x³/9 + C. The option with x³/3 forgets to divide by 3 again.

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