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\title{Math 242, Lecture 4}
\maketitle

\section{ Integration by substitution} 

While we might not have thought about it, 
the chain rule was by far the rule we used most
frequently in finding derivatives.    
In the world of integration, the cousin of the
chain rule is the technique of ``u-substitution''
for integration.  We make steps similar to 
the steps we took when applying the chain rule.  
To evaluate $\int f(x) dx$, 
\begin{itemize}
\item First we identify a function traditionally called
$u(x)$   (hence the name ``u-substitution'')   
which we use as a building block to make the integrand.  
In particular, we want to have some $g(u(x))$ appearing
naturally as part of the integrand.  
\item Then, we take the derivative of $u(x)$,  
which is part of the chain rule.  
\item We substitute the variable $u$ in for $u(x)$   (this is
easy)   and then - here's the rub! -   try to match what's remaining with
$du$   (which is $\frac{du}{dx} dx$).    In this step we often use 
the trick of multiplying by some number $k$ inside the integral and
then $\frac{1}{k}$ outside the integral.  
\item If the substitution meshes, we have successfully translated the
integral $\int f(x) dx = \int g(u) du$.  
\item If we can integrate $\int g(u) du = G(u) + C$   then 
we can substitute $u(x)$ for $u$ to find the original integral.  
\end{itemize}

The only way to learn this technique is through  plenty of practice.  
(It's a bit of an art).  

\begin{example}
Evaluate the following integrals:
\begin{itemize}
\item $\int (x^4 + 2x - 1)^{-5} (2x^3 + 1) dx$  
\item $\int e^{x^3 - 1} x^2 dx$  
\item $\int \frac{1}{x^2 - 4} x dx$  
\item $\int \frac{1}{x \ln x} dx$  
\item $\int \frac{x^2 + 2}{x + 1} dx$  
\item $\int \frac{e^x}{e^x + 1} dx$  
\item $\int e^{-x^2} dx$  
\end{itemize}
\end{example}

\subsection{ Tips and more practice}  

A few tips to keep in mind while doing integration problems:  
\begin{itemize}
\item It is crucial to substitute in $du$ as well as $u$, and not have
any expressions involving $x$ left when the substitution is made.  
\item Most $u$ substitutions will not result in a manageable integral;  
it takes some practice to know which substitutions have the greatest chance
of success.   (In fact, it can be helpful to see some ``dead-end'' substitutions.)  
\item In fact, not all indefinite integrals are computable explicitly.  
\end{itemize}

\begin{example}
Evaluate the following integrals:
\begin{itemize}
\item $\int x^2 \sqrt{x^3 + 5} dx$ 
\item $\int \frac{e^x}{(e^x + 5)^3} dx$ 
\item $\int \frac{x}{x+1} dx$  
\end{itemize}
\end{example}



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