What is the integral of 2x+1/y-y/(x^2) ?

The integral of 2x+1/y-y/(x^2) is x/y+y/x+x^2+C

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integral of 2x+1/y-y/(x^2)

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

\frac{x}{y} + \frac{y}{x} + x^{2} + C

Solution steps

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

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

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

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

= \frac{x}{y} - (- \frac{y}{x}) + x^{2}

Simplify

= \frac{x}{y} + \frac{y}{x} + x^{2}

Add a constant to the solution

= \frac{x}{y} + \frac{y}{x} + x^{2} + C

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What is the integral of 2x+1/y-y/(x^2) ?
The integral of 2x+1/y-y/(x^2) is x/y+y/x+x^2+C