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CA Foundation · Quantitative Aptitude · Differential and Integral Calculus

Evaluate ∫ x e^x dx (C is the constant of integration).

The integral equals x e^x − e^x + C. Using integration by parts with u = x and dv = e^x dx, the integral becomes x e^x minus the integral of e^x, which is e^x. Differentiating the result returns x e^x, confirming it.

  1. Ax e^x + e^x + C
  2. Bx e^x − e^x + CCorrect
  3. C(x²/2) e^x + C
  4. Dx e^x − x + C

Explanation

Take u = x and dv = e^x dx. Then ∫x e^x dx = x e^x − ∫e^x dx = x e^x − e^x + C. Differentiating gives e^x + x e^x − e^x = x e^x, which confirms the result. The option with +e^x comes from a sign error in the parts formula.

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