What is the integral of (x+(3y)/x) ?

The integral of (x+(3y)/x) is 3yln|x|+(x^2)/2+C

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integral of (x+(3y)/x)

\int (x + \frac{3 y}{x}) d x

3 y ln ∣ x ∣ + \frac{x ^{2}}{2} + C

Solution steps

\int x + \frac{3 y}{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 \frac{3 y}{x} d x + \int x d x

\int \frac{3 y}{x} d x = 3 y ln ∣ x ∣

\int x d x = \frac{x ^{2}}{2}

= 3 y ln ∣ x ∣ + \frac{x ^{2}}{2}

Add a constant to the solution

= 3 y ln ∣ x ∣ + \frac{x ^{2}}{2} + C

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What is the integral of (x+(3y)/x) ?
The integral of (x+(3y)/x) is 3yln|x|+(x^2)/2+C