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CA Foundation · Quantitative Aptitude · Differential and Integral Calculus

Evaluate the definite integral ∫ from 1 to 2 of (3/x + 2e^(2x)) dx, taking the value in terms of e and log.

The value is 3 log 2 + e⁴ − e². The term 3/x integrates to 3 log x, which gives 3 log 2 between 1 and 2. The term 2e^(2x) integrates to e^(2x) because dividing by the inner derivative 2 cancels the coefficient, giving e⁴ − e².

  1. A3 log 2 + e⁴ − e²Correct
  2. B3 log 2 + 2e⁴ − 2e²
  3. C3 log 2 + e⁴ + e²
  4. D3 log 2 + 4e⁴ − 4e²

Explanation

∫3/x dx = 3 log x, giving 3 log 2 − 3 log 1 = 3 log 2. ∫2e^(2x)dx = e^(2x), giving e⁴ − e². Total is 3 log 2 + e⁴ − e². The option with 2e⁴ − 2e² forgets the factor 1/2 from the chain-rule reversal.

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