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

Solución

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

Solución

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

Pasos de solución

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

Aplicar integración por sustitución

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

Sacar la constante:

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

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

Factorizar

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

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

Sacar la constante:

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

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

Simplificar

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

Tomar la fracción parcial de

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

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

Aplicar la regla de la suma:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Sustituir en la ecuación

u = \frac{y}{x}

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

Simplificar

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

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

Agregar una constante a la solución

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

Ejemplos populares

derivative e^{xy^2}derivada de

e^{x y^{2}}

derivative (2x)/(-x+2)derivada de

\frac{2 x}{- x + 2}

integral de 1/(3^{2x)}

\int \frac{1}{3 ^{2 x}} d x

derivada de-1/(2x^{3/2})

\frac{d}{d x} (- \frac{1}{2 x ^{\frac{3}{2}}})

límite cuando x tiende a 0 de tan^x(2x)

x \to 0 lim (tan^{x} (2 x))