What is the integral of (e^x+ln(y)+y/x) ?

The integral of (e^x+ln(y)+y/x) is yln|x|+ln(y)x+e^x+C

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integral of (e^x+ln(y)+y/x)

\int (e^{x} + ln (y) + \frac{y}{x}) d x

y ln ∣ x ∣ + ln (y) x + e^{x} + C

Solution steps

\int e^{x} + ln (y) + \frac{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{y}{x} d x + \int ln (y) d x + \int e^{x} d x

\int \frac{y}{x} d x = y ln ∣ x ∣

\int ln (y) d x = ln (y) x

\int e^{x} d x = e^{x}

= y ln ∣ x ∣ + ln (y) x + e^{x}

Add a constant to the solution

= y ln ∣ x ∣ + ln (y) x + e^{x} + C

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