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CA Foundation · Quantitative Aptitude · Differential and Integral Calculus

The indefinite integral ∫ x·eˣ dx is equal to (where C is the constant of integration):

Using integration by parts with u = x and dv = eˣ dx, the integral equals x·eˣ − eˣ + C, which is (x − 1)eˣ + C. Differentiating this result gives x·eˣ, confirming it is the correct antiderivative.

  1. A(x − 1)eˣ + CCorrect
  2. B(x + 1)eˣ + C
  3. Cx·eˣ + C
  4. Dx²eˣ/2 + C

Explanation

Take u = x and dv = eˣ dx. Then ∫x eˣ dx = x eˣ − ∫eˣ dx = x eˣ − eˣ + C = (x − 1)eˣ + C. Option (x + 1)eˣ + C results from adding instead of subtracting the second integral; its derivative is (x + 2)eˣ, which is not x eˣ.

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