YLL討論網

∫1 /(1+X^2) ^2dX at((0, ∞)
令X=tan Y, dX/dY=sec^2Y  ,, Y at (0, π/2)
∫1 /(1+X^2) ^2dX=∫1 /(1+tan^2Y) ^2 *sec^2Y dY =∫1 /sec^2Y dY=
∫cos^2Y dY at (0, π/2)
=∫(1+cos2Y)/2 dY=1/2∫(1+cos2Y) dY=1/2(Y| at (0, π/2)+ (1/2)sin2Y| at (0, π/2))
=π/4

令X=tan Y,
∫X^2 /(1+X^2) ^2dX=∫tan^2Y /(1+tan^2Y) ^2 *sec^2Y dY =
∫tan^2Y /sec^2Y dY=∫sin^2Y dY=∫cos^2Y dY at (0, π/2)

算法精簡, 但是很難理解.
請問在原因中Fs(w), Gs(w), f(t), g(t)各代表什麼?

This is trivial.

Make the substitution x -> 1/x we get the first part.

It suffices to consider int_0^infinity 1/(1+x^2) dx = arctanx | 0 -> infinity = pi/2

so required integral = 1/2 * pi/2 = pi/4



Why should one use advanced method for a trivial question?

It's nice for x→1/x to get the first part.
But can't understand why the required integral [∫1 /(1+x^2) ^2dx] equal to 1/2 of ∫1/(1+x^2) dx?

If they are equal, sum them up!