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\centerline{MATH 136 -- Calculus 2}
\centerline{Second Practice Day on $u$-substitution}
\centerline{February 5, 2020}
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\noindent
{\it Background}
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Here are a few additional integration formulas that can be used
in conjunction with $u$-substitution:
\begin{itemize}
\item If $b > 0$, then $\displaystyle \int b^x\ dx = \frac{b^x}{\ln(b)} + C$
(This follows by combining the derivative rule for $e^u$ by the chain rule 
with the formula $b^x = (e^{\ln(b)})^x = e^{x\ln(b)}$.)
\item $\displaystyle \int \frac{dx}{|x|\sqrt{x^2 - 1}} = \sec^{-1}(x) + C$
(follows from the derivative rule 
$$\frac{d}{dx}\sec^{-1}(x) = \frac{1}{|x|\sqrt{x^2 - 1}}$$
\end{itemize}
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{\it Questions}
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\noindent
Find the following integrals using the formulas above and $u$-substitution as needed: 
\begin{enumerate}
\item $\displaystyle \int 5^{\cos(x)} \sin(x)\ dx$
\item $\displaystyle \int \frac{dx}{(x + 8)\ln(2x + 16)}$
\item $\displaystyle \int \frac{dx}{x \sqrt{49x^2 - 1}}$ -- take $(x > 0)$
\item $\displaystyle \int x^2\sqrt{x + 4}\ dx$ (Hint:  Let $u = x + 4$)
\item $\displaystyle \int_0^1 4^{3x}\ dx$ 
\end{enumerate}

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