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CA Foundation · Quantitative Aptitude · Differential and Integral Calculus

Evaluate ∫ (e^(3x) + 4/x) dx for x > 0.

The integral is (1/3)e^(3x) + 4 ln x + c. The exponential term needs division by the coefficient 3 of x in the exponent, and 4/x integrates to 4 times the natural log of x.

  1. Ae^(3x) + 4 ln x + c
  2. B3e^(3x) + 4 ln x + c
  3. C(1/3)e^(3x) + 4 ln x + cCorrect
  4. D(1/3)e^(3x) − 4/x² + c

Explanation

∫e^(3x) dx = e^(3x)/3 because d/dx(e^(3x)/3) = e^(3x). ∫4/x dx = 4 ln x. So the result is (1/3)e^(3x) + 4 ln x + c. Omitting the 1/3 factor, as in the first option, ignores the chain rule in reverse.

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