MHT CET202620 April 2026Evening ShiftMathematicsDefinite IntegrationActual
The value of the integral _ 1/e ^ e | x| x^2 dx is
Options
- Ae^2 - 1 2e
- B2 e
- C2 - 2 e
- D1 - 1 e
Correct answer
C. 2 - 2 e
Step-by-step solution
Let I = _ 1/e ^ e | x| x^2 dx Substitute x = e^t dx = e^t dt When x = 1/e , t = -1 and when x = e , t = 1 I = _ -1 ¹ |t| e^ 2t e^t dt = _ -1 ¹ |t| e^ -t dt Splitting the integral at t = 0 : I = _ -1 ⁰ -t e^ -t dt + ₀¹ t e^ -t dt Using integration by parts: -t e^ -t dt = t e^ -t + e^ -t t e^ -t dt = -t e^ -t - e^ -t Evaluating the definite integrals: I = [ t e^ -t + e^ -t ]_ -1 ⁰ + [ -t e^ -t - e^ -t ]₀¹ I = ( (0 + e^0) - (-e^1 + e^1) ) + ( (-e⁻¹ - e⁻¹) - (0 - e^0) ) I = (1 - 0) + ( - 2 e - (-1) ) I = 1 + 1 - 2 e =