What is the integral of y^{-1}+2/x ?

The integral of y^{-1}+2/x is 2ln|x|+x/y+C

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integral of y^{-1}+2/x

\int y^{- 1} + \frac{2}{x} d x

2 ln ∣ x ∣ + \frac{x}{y} + C

Solution steps

\int y^{- 1} + \frac{2}{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{2}{x} d x + \int y^{- 1} d x

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int y^{- 1} d x = \frac{x}{y}

= 2 ln ∣ x ∣ + \frac{x}{y}

Add a constant to the solution

= 2 ln ∣ x ∣ + \frac{x}{y} + C