Question

can you help me find
integration [2xy/(x^2+y^2)^2] do x
?

Answer 1

∫ 2 x y dx / ( x² + y² )² =

2 y ∙ ∫ x ∙ dx / ( x² + y² )² =
_______________
Apply substitution:

u = x² + y²
______________

2 y ∙ ∫ x ∙ dx / u² =

____________________________
Derivative:

du / dx = d ( x² ) / dx + d ( y² ) / dx

du / dx = 2 x + 0

du / dx = 2 x

du = 2 x dx

2 x dx = du

x dx = du / 2
____________________________

= 2 y ∙ ∫ ( du / 2 ) / u² =

= ( 2 y / 2 ) ∙ ∫ du / u² =

= y ∙ ∫ du / u² =

y ∙ ∫ u⁻² ∙ du / =

y ∙ u ⁻ ² ⁺ ¹ / ( - 2 + 1 ) + C

y ∙ u ⁻ ¹ / ( - 1 ) + C =

y ∙ ( - 1 / u ) + C =

- y / u + C =

- y / ( x² + y² ) + C