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	{\noindent 21-120 Differential and Integral Calculus \hfill Summer Session I 2010\\}
	{\centering Quiz 11\medskip\\}
	\hrule
	{
		\begin{prob*} \mbox{}
			\begin{description}
				\item[1.] $\ds\int\left(x+\frac{1}{x}\right)dx$
				\item[2.] $\ds\int\left(x^3+3x^2+2x+\frac{1}{3}\right)dx$
				\item[Bonus.] $\ds\int\sin(\frac{x}{2})\cos(\frac{x}{2})\;dx$
			\end{description}
		\end{prob*}
		\begin{soln*}
			\mbox{}
			\begin{description}
				\item[1.]
				\begin{align*}
					\int\left(x+\frac{1}{x}\right)dx 	&= \left(\int x\;dx\right) +\left(\int\frac{1}{x}\;dx\right) \\
										&= \frac{x^2}{2} + \ln|x| + C
				\end{align*}
				\item[2.]
				\begin{align*}
					\int\left(x^3+3x^2+2x+\frac{1}{3}\right)dx	&= \left(\int x^3\;dx\right) + \left(3\int x^2\;dx\right) + \left(2\int x\;dx\right) + \left(\frac{1}{3}\int dx\right) \\
												&= \frac{x^4}{4} + x^3 + x^2 + \frac{x}{3} + C
				\end{align*}
				\item[Bonus.] Recall, $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$. So,
				\begin{align*}
					\int\sin(\frac{x}{2})\cos(\frac{x}{2})\;dx 	&= \int \frac{\sin(x)}{2} \\
												&= \frac{1}{2} \int \sin(x) \\
												&= -\frac{cos(x)}{2}
				\end{align*}


			\end{description}

		\end{soln*}

	}
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