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

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

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

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

- \frac{1}{x} - \frac{y ^{2}}{x} + x + C

Solution steps

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

Expand

\frac{1 + y ^{2} + x ^{2}}{x ^{2}} : \frac{1}{x ^{2}} + \frac{y ^{2}}{x ^{2}} + 1

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

\int \frac{1}{x ^{2}} d x = - \frac{1}{x}

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

\int 1 d x = x

= - \frac{1}{x} - \frac{y ^{2}}{x} + x

Add a constant to the solution

= - \frac{1}{x} - \frac{y ^{2}}{x} + x + C

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