What is the integral of (e^{3x}+1/x) ?

The integral of (e^{3x}+1/x) is 1/3 e^{3x}+ln|x|+C

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integral of (e^{3x}+1/x)

\int (e^{3 x} + \frac{1}{x}) d x

\frac{1}{3} e^{3 x} + ln ∣ x ∣ + C

Solution steps

\int e^{3 x} + \frac{1}{x} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int e^{3 x} d x + \int \frac{1}{x} d x

\int e^{3 x} d x = \frac{1}{3} e^{3 x}

\int \frac{1}{x} d x = ln ∣ x ∣

= \frac{1}{3} e^{3 x} + ln ∣ x ∣

Add a constant to the solution

= \frac{1}{3} e^{3 x} + ln ∣ x ∣ + C

x \to 0 lim (\frac{2 ^{3 x} - 1}{x})

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What is the integral of (e^{3x}+1/x) ?
The integral of (e^{3x}+1/x) is 1/3 e^{3x}+ln|x|+C