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\centerline{MATH 136 -- Calculus 2}
\centerline{Worksheet on Antiderivatives}
\centerline{September 7, 2016}
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\noindent
{\it Background}
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As we have just seen, the \emph{indefinite integral} (note -- no limits of integration on 
the integral sign here; that is the ``tip-off'' that we're not talking about 
a definite integral, e.g. for an area):
$$F(x) = \int f(x)\ dx$$
is shorthand notation for \emph{the general antiderivative} -- that means the most
general function such that $F'(x) = f(x)$.  For instance as we know the antiderivative
Power Rule:
$$\int x^n\ dx = \begin{cases} \displaystyle{\frac{x^{n+1}}{n+1} + C} & \text{ if } n \ne -1, \\
                               \displaystyle{\ln|x| + C} & \text{ if } n = -1.
                 \end{cases}
$$
We want to review some material from
differential calculus by ``thinking backwards'' for the rest of the time today.
Find the following anti
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\noindent
{\it Questions}
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\begin{enumerate}
\item[1)]  Find the following antiderivatives (indefinite integrals):
\begin{enumerate}
\item[(a)] $\displaystyle{\int \sqrt{x} + \frac{1}{\sqrt[3]{x}} + \frac{4}{x^3}\ dx}$ 
(Hint:  Write as powers and use the Power Rule above)
\item[(b)] $\displaystyle{\int \cos(x/2)\ dx}$  (``guess and check'')
\item[(c)] $\displaystyle{\int e^{5x}\ dx}$ (``guess and check'')
\item[(d)] $\displaystyle{\int \frac{1}{\sqrt{1 - x^2}}\ dx}$  (Hint: What function has a 
derivative that looks like this?  It's not the inverse tangent that we discussed.)
\end{enumerate}
\item[2)] How do we know that \emph{every} antiderivative of a function like $f(x) = x^2$
looks like $F(x) = \frac{x^3}{3} + C$?  They certainly all ``work,'' but could there be
\emph{others too}?  
\begin{enumerate}
\item[(a)] Suppose you have the antiderivative $F_1(x) = \frac{x^3}{3}$ and $F_2(x)$ is 
\emph{any other antiderivative} of $x^2$.  What do you get if you take the derivative of the 
difference $F_2(x) - F_1(x)$?
\item[(b)]  What does that say about the difference $F_2(x) - F_1(x)$?
\item[(c)]  So what does $F_2(x)$ look like? 
\end{enumerate}
\end{enumerate}
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