What is the integral of 1+ln(x)+y/x ?

The integral of 1+ln(x)+y/x is xln(x)+yln|x|+C

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integral of 1+ln(x)+y/x

\int 1 + ln (x) + \frac{y}{x} d x

x ln (x) + y ln ∣ x ∣ + C

Solution steps

\int 1 + ln (x) + \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 1 d x + \int \frac{y}{x} d x + \int ln (x) d x

\int 1 d x = x

\int \frac{y}{x} d x = y ln ∣ x ∣

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

= x + y ln ∣ x ∣ + x ln (x) - x

Simplify

= x ln (x) + y ln ∣ x ∣

Add a constant to the solution

= x ln (x) + y ln ∣ x ∣ + C

What is the integral of 1+ln(x)+y/x ?
The integral of 1+ln(x)+y/x is xln(x)+yln|x|+C