Hi guys i've got a problem with an integral question:
'integral sign' t/(2^(2t)). Can someone please help me with this. I dont know what to do. Thanks in advance
Hi guys i've got a problem with an integral question:
'integral sign' t/(2^(2t)). Can someone please help me with this. I dont know what to do. Thanks in advance
Working from here, we have $\displaystyle u=t$ and $\displaystyle dv=e^{at}\,dt$. Thus $\displaystyle du = dt$ and $\displaystyle v = \frac{1}{a}e^{at}$. Now we have:
$\displaystyle \frac{t}{a}e^{at}-\frac{1}{a}\int e^{at} = \frac{t}{a}e^{at}-\frac{1}{a^2}e^{at} = \frac{e^{at}}{a^2}(at-1)$
Since $\displaystyle a=-2\ln(2)$ and $\displaystyle e^{-2t\ln(2)} = 2^{-2t}$, we have:
$\displaystyle \frac{e^{at}}{a^2}(at-1) = \frac{2^{-2t}(-2t\ln(2)-1)}{4(\ln(2))^2} = \boxed{-\frac{2^{-2t}(2t\ln(2)+1)}{4(\ln(2))^2}}$