How to determine the function f(x) if f'(x)=2e^ln(2x+3)/(2x+3)?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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To determine the primitive function f(x), we must calculate the indefinite integral of f'(x).

We'll apply substitution technique, replacing ln(2x+3) by t.

ln(2x+3) = t

We'll differentiate both sides and we'll get:

2dx/(2x+3) = dt

We'll re-write the integral in the new variable:

Int 2e^(ln(2x+3)) dx/(2x+3) = Int e^t*dt

Int e^t*dt = e^t + C

Int 2e^(ln(2x+3)) dx/(2x+3) = e^(ln(2x+3)) + C

The requested primitive function is: f(x) = e^(ln(2x+3)) + C

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