Simple Integration by Substitution

[Tracker]

I am just doing a simple integral using substitution and when I check it by taking the derivative it not coming out as the same answer.

 

I am taking [math] \int \frac{x}{(2+x^2)^2} [/math] and getting [math] \frac{-1}{2(2+x^2)} [/math]. When I take the derivative of that function I get [math] \frac{4x}{(4+2x^2)^2} [/math]. Where is my mistake?

[Cap'n Refsmmat]

Your integration seems correct. As for the deriving...

 

[math]\frac{d}{dx} \frac{-1}{2(2 + x^2)} = \frac{d}{dx} -1 (4 + 2x^2)^{-1}[/math]

 

[math]= (4 + 2x^2)^{-2} (4x)[/math]

 

[math]= \frac{4x}{(4 + 2x^2)^2}[/math]

 

[math]= \frac{4x}{16 + 16x^2 + 4x^4}[/math]

 

[math]= \frac{4x}{4(4 + 4x^2 + x^4)}[/math]

 

[math]= \frac{4x}{4(2 + x^2)^2}[/math]

 

[math]= \frac{x}{(2 + x^2)^2}[/math]

 

Poof. You had it right all along.