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\centerline{MATH 134 -- Calculus with Fundamentals 2}
\centerline{Another Practice Day on $u$-substitution}
\centerline{February 12, 2018}
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\noindent
{\it Background}
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Today, we want to continue working on $u$-substitutions
involving other transcendental functions.
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\noindent
{\it Questions}

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For each problem, 
\begin{enumerate}
\item[(i)] find a candidate $u$,
\item[(ii)]  compute $du = \frac{du}{dx}\ dx$
\item[(iii)]  see whether the rest of the integrand can be matched with $du$, 
possibly up to a constant multiple (if not, then you might need to try a 
different $u$), 
\item[(iv)] finish the integration.
\end{enumerate}
\begin{enumerate}
\item $\displaystyle \int_1^{\sqrt{3}} \frac{dx}{\tan^{-1}(x)(1 + x^2)}\ dx$
\item $\displaystyle \int_e^{e^2} \frac{dx}{x (\ln(x))^4}\ dx$
\item $\displaystyle \int \frac{\ln(\ln(x))}{x\ln(x)}\ dx$  (Note:  There are several possible $u$
to try here; keep going until you find the right one!) 
\item $\displaystyle \int e^t\sqrt{e^t - 1}\ dt$ 
\item $\displaystyle \int 8^x\ dx$ (Hint: rewrite $8^x$ as $8^x = (e^{\ln(8)})^x = e^{x\ln(8)}$, 
then think how you would do this by a $u$-substitution.
The same trick will work to integrate any $\int b^x\ dx$ where $b > 0$.  See page 315 in 
Rogawski and Adams.)
\end{enumerate}

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