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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{2 u ^{2}} d u

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

Take the constant out:

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

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

Apply the Power Rule

: - \frac{1}{u}

= - 2 x \frac{1}{2} (- \frac{1}{u})

Substitute back

u = x^{2} + y^{2}

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

Simplify

- 2 x \frac{1}{2} (- \frac{1}{x ^{2} + y ^{2}}) : \frac{x}{x ^{2} + y ^{2}}

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

Add a constant to the solution

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

x \to - 3 lim (3 x + 2)

\frac{\partial}{\partial x} (y cos (x) + sin (x))

\int x^{2} (6 - x^{3})^{5} d x

(- 3) (1.2)

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

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