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\flushleft\underline{Homework 3 --- MATH 1586 Spring 2018} \\
Recall the technique of integration by parts is of the form
\[\displaystyle\int u \mathrm{d}v = uv - \displaystyle\int v \mathrm{d}u.\] 
Also recall that we derived the antiderivative of the natural logarithm using integration by parts:
\[ \displaystyle\int \ln(x) \mathrm{d}x = x\ln(x)-x + C.\]
Recall that an ``improper integral" of the form $\displaystyle\int_a^{\infty}$ or of the form $\displaystyle\int_{-\infty}^b$ is understood in the following way:
\[ \displaystyle\int_a^{\infty} f(x) \mathrm{d}x = \displaystyle\lim_{b \rightarrow \infty} \displaystyle\int_a^b f(x) \mathrm{d}x.\]
and
\[ \displaystyle\int_{-\infty}^b f(x) \mathrm{d}x = \displaystyle\lim_{a \rightarrow -\infty} \displaystyle\int_a^b f(x) \mathrm{d}x.\]
Finally, recall the definition of the Laplace transform of a function $f$:
$$\mathscr{L}\{f\}(x)=\displaystyle\int_0^{\infty} f(t)e^{-xt} \mathrm{d}t.$$
\begin{enumerate}
\item (``RL-Circuit problem")\\
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  \begin{center}
    \begin{circuitikz}
      \draw (0,0)
      to[V,v=$E$] (0,2) % The voltage source
      to[L=$L$] (2,2)
      to[short] (2,0) % The resistor
      to[R=$R$] (0,0);
    \end{circuitikz}
  \end{center}
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 In a series circuit containing only a resistor ($R$, measured in ohms), an inductor ($L$, measured in henries which are $\mathrm{ohm}\cdot \mathrm{sec}$), and a current at time $t$, $i(t)$ (measured in amps), Kirchoff's second law states that the sum of the voltage drop across the inductor $\left( L \dfrac{\mathrm{d}i}{\mathrm{d}t} \right)$ and the voltage drop across the resistor ($iR$) is the same as the impressed voltage ($E(t)$) on the circuit. We obtain the differential equation for the current $i(t)$
$$L \dfrac{\mathrm{d}i}{\mathrm{d}t} + Ri(t) = E(t),$$
where $L$ and $R$ are constants known as the inductance and resistance.

A 25-volt electromotive force is applied to a series circuit in which the inductance is $1$ henry and the resistance is $30$ ohms.
$i' + 30i = 25$

\begin{enumerate}[a.)]
\item What is the differential equation we must solve here?
\item Calculate $\dfrac{\mathrm{d}}{\mathrm{d}t} \Big[ e^{30t}i(t) \Big]$. What do you notice about this compared to the left-hand side of your differential equation?
\item  Multiply your differential equation on both sides by $e^{30t}$ and then rewrite the left-hand side as $\dfrac{\mathrm{d}}{\mathrm{d}t} \Big[ e^{30t}i(t) \Big]$.
\item Solve the differential equation by integrating and solving for $i(t)$. 
\end{enumerate}

\item Compute $\displaystyle\int_5^{11} xe^x \mathrm{d}x$.

\item Compute $\displaystyle\int x^2 e^x \mathrm{d}x$.

\item Compute $\displaystyle\int (2x+3)e^{2x} \mathrm{d}x$.

\item Compute $\displaystyle\int_2^3 \ln(x) \mathrm{d}x$.

\item Let $f(t)=e^{7t}$. 
\begin{enumerate}
\item What integral must you solve in order to calculate $\mathscr{L}\{f\}(x)$?
\item Calculate $\mathscr{L}\{f\}(x)$ as an improper integral.
\end{enumerate}

\item Let $f(t)=t$. 
\begin{enumerate}
\item Let $x>0$. What integral must you solve in order to calculate $\mathscr{L}\{f\}(x)$?
\item Calculate $\mathscr{L}\{f\}(x)$ using integration by parts and improper integration.
\end{enumerate}

\end{enumerate}
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