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

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

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

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

- arctan (\frac{y}{x}) + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integral Substitution

= - x \cdot \int \frac{1}{x ( u ^{2} + 1 )} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Use the common integral:

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

= - x \frac{1}{x} arctan (u)

Substitute back

u = \frac{y}{x}

= - x \frac{1}{x} arctan (\frac{y}{x})

Simplify

- x \frac{1}{x} arctan (\frac{y}{x}) : - arctan (\frac{y}{x})

= - arctan (\frac{y}{x})

Add a constant to the solution

= - arctan (\frac{y}{x}) + C

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