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Now, I will do some manipulations to get a different expression: ##\displaystyle f(t) = \int_0^{\infty} \frac{-\sin (tx) x}{x^2+1}~ dx = \int_0^{\infty} \frac{-\sin (tx) x^2}{(x^2+1)x}~ dx = \int_0^{\infty} \frac{-\sin (tx) (x^2+1) + \sin (tx)}{(x^2+1)x}~ dx = \int_0^{\infty} \frac{- \sin(tx)}{x} ~ dx + \int_0^{\infty} \frac{\sin (tx)}{(x^2+1)x}~ dx = -\frac{\pi}{2} + \int_0^{\infty} \frac{\sin (tx)}{(x^2+1)x}~ dx##. So here, we see that ##f(0) = - \frac{\pi}{2}##.

What am I doing wrong? Why am I getting two different values?