Compute the Energy and Power of the signal $ x(t)=\dfrac{2t}{t^2+5} $ between 0 and 2 seconds.
Energy
$ E=\int_0^{2}{\dfrac{2t}{t^2+5}dt} $
$ U=t^2+5 $
$ dU=2tdt $
Limits:
$ U(0)=0^2+5=5 $
$ U(2)=2^2+5=4+5=9 $
$ E=\int_{5}^{9}\dfrac{du}{U} $
$ E=\ln U |_{U=5}^{U=9} $
$ E=\ln 9 - \ln 5 $
$ E=\ln{(\dfrac{9}{5})} $
Power
$ U=\int_{t_1}^{t_2}x(t)dt $
$ U=\int_0^{2}{\dfrac{2t}{t^2+5}dt} $
Limits:
$ U(0)=0^2+5=5 $
$ U(2)=2^2+5=9 $
$ P=\dfrac{1}{2}\int_{5}^{9}\dfrac{du}{U} $
$ P=\dfrac{1}{2}\ln U |_{U=5}^{U=9} $
$ P=\dfrac{1}{2}(\ln 9 - \ln 5) $
$ P=\dfrac{1}{2}\ln{(\dfrac{9}{5})} $
