CA Foundation · Quantitative Aptitude · Differential and Integral Calculus
The value of the definite integral of x·e^(2x) with respect to x from 0 to 1 is:
The integral equals (e² + 1)/4. Integration by parts gives x·e^(2x)/2 − e^(2x)/4. Evaluating at 1 gives e²/4 and at 0 gives −1/4. Subtracting the lower value adds 1/4. A sign slip at the lower limit would give (e² − 1)/4.
- A(e² − 1)/4
- B(e² + 1)/4Correct
- Ce²/4
- D(e² + 1)/2
Explanation
By parts, ∫x e^(2x)dx = x·e^(2x)/2 − ∫e^(2x)/2 dx = x e^(2x)/2 − e^(2x)/4. At x = 1 this is e²/2 − e²/4 = e²/4. At x = 0 it is −1/4. The difference is e²/4 + 1/4 = (e² + 1)/4; the option (e² − 1)/4 comes from a sign error at the lower limit.
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