EE102 Systems and Signals (Discussion)

Course description
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Examples

1. $\int t e^{at} dt$
   Solution: In the usual way, we identify
   $$u = t, dv = e^{at} dt \Longrightarrow du = dt, v = \frac{e^{at}}{a}$$
   Therefore, integrating by parts,
   

   $$\int t e^{at} dt = t \frac{e^{at}}{a} - \int \frac{e^{at}}{a} dt$$

   $$= \frac{t e^{at}}{a} - \frac{e^{at}}{a^2} + C$$

   
   
   where $C$ is an arbitrary constant, usually included in indefinite integrals.

2. $\int_0^\infty t^n e^{-t} dt$
   Solution: Let
   $$I_n = \int_0^\infty t^n e^{-t} dt$$
      and let $$u = t^n, dv = e^{-t} dt \Longrightarrow du = n t^{n-1} dt, v = -e^{-t}$$
   Therefore, integrating by parts, we get
   

   $$I_n = \left[ t^n(-e^{-t}) \right]_0^\infty - \int_0^\infty (-e^{-t}) n t^{n-1} dt$$

   $$= 0 + n \int_0^\infty t^{n-1} e^{-t} dt$$

   $$= n I_{n-1}, ~n \geq 1.$$

   
   
   So,
   

   $$I_1 = 1 \cdot I_{0} = I_0$$

   $$I_2 = 2\cdot I_1 = 2\cdot 1 \cdot I_0$$

   $$I_3 = 3\cdot I_1 = 3 \cdot 2\cdot 1 \cdot I_0$$

   
   
   and in general,
   $$I_n = n(n-1)...1 \cdot I_0 = (n!) \cdot I_0$$
   Finally, since
   $$I_0 = \int_0^\infty t^0 e^{-t} dt = 1$$
   we get
   $$I_n =(n!),~n \geq 1$$

   Actually $I_n$ can be defined for non-integral values of $n$, and is a generalization of the factorial to non-integral $n$. $\Gamma(n) = I_{n-1}$ is called the gamma function.

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