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calculus:resources:calculus_flipped_resources:applications:4.5_substitution_tex

TeX code compiled with \documentclass{beamer} using the Amsterdam theme.

\begin{document} \begin{frame} The differentiation rule that helps us understand why the Substitution rule works is: \vskip 20pt \begin{enumerate}[a)] \item The product rule. \vskip 10pt \item The chain rule. \vskip 10pt \item The quotient rule. \vskip 10pt \item All of the above. \end{enumerate} \end{frame} \begin{frame} Find the indefinite integrals. \begin{columns} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf (i)] $\displaystyle\int x^2\sqrt{x^3+21}\,dx$ \vskip 20pt \item[\bf (ii)] $\displaystyle\int \cos^4(\theta)\sin(\theta)\,d\theta$ \vskip 20pt \item[\bf (iii)] $\displaystyle\int (9t+7)^{2.5}\,dt$ \end{itemize} \end{column} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf (iv)] $\displaystyle\int (x+5)\sqrt{10x+x^2}\,dx$ \vskip 20pt \item[\bf (v)] $\displaystyle\int \frac{z^3}{\sqrt[3]{3+z^4}}\,dz$ \vskip 20pt \item[\bf (vi)] $\displaystyle\int x(8x+7)^8\,dx$ \end{itemize} \end{column} \end{columns} \end{frame} \begin{frame} Find the indefinite integrals and evaluate the definite integrals. \begin{columns} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf (i)] $\displaystyle\int x^3\sqrt{x^2+4}\,dx$ \vskip 20pt \item[\bf (ii)] $\displaystyle\int x^5\sin(x^6)\,dx$ \vskip 20pt \item[\bf (iii)] $\displaystyle\int \sec^2(\theta)\tan^7(\theta)\,d\theta$ \end{itemize} \end{column} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf (iv)] $\displaystyle\int \sqrt{x^5}\sin(2+x^{7/2})\,dx$ \vskip 20pt \item[\bf (v)] $\displaystyle\int \frac{\cos(\pi/x^{29})}{x^{30}}\,dx$ \vskip 20pt \item[\bf (vi)] $\displaystyle\int \sin(45t)\sec^2(\cos(45t))\,dt$ \end{itemize} \end{column} \end{columns} \end{frame} \begin{frame} If $f$ is continuous and $\displaystyle\int_0^4 f(x)\,dx=2$, find $\displaystyle\int_0^2 f(2x)\,dx$. \end{frame} \begin{frame} Evaluate the definite integrals. \begin{columns} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf (i)] $\displaystyle\int_0^1 \sqrt[3]{1+7x}\,dx$ \vskip 20pt \item[\bf (ii)] $\displaystyle\int_0^{\sqrt[14]{\pi}} x^{13}\cos(x^{14})\,dx$ \vskip 20pt \item[\bf (iii)] $\displaystyle\int_0^{\pi/10} \cos(5x)\sin(\sin(5x))\,dx$ \end{itemize} \end{column} \begin{column}{0.5\textwidth} \begin{itemize} \item[\bf (iv)] $\displaystyle\int_0^{31}\frac{dx}{\sqrt[3]{(1+4x)^2}}$ \vskip 20pt \item[\bf (v)] $\displaystyle\int_9^{10} x\sqrt{x-9}\,dx$ \end{itemize} \end{column} \end{columns} \end{frame} \end{document}