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\centerline{MATH 134 -- Calculus with Fundamentals 2}
\centerline{Practice Day on Antiderivatives}
\centerline{February 5, 2018}
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\noindent
{\it Background}
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If $f(x)$ is a given function and $F(x)$ is another function that satisfies
$F'(x) = f(x)$, then we say $F(x)$ is an \emph{antiderivative} of $f(x)$.
Today we are going to practice on finding antiderivatives of functions defined
by formulas by using our basic derivative rules, but ``in reverse.''
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\noindent
{\it Questions}

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\begin{enumerate}
\item[(1)] Find an antiderivative for $f(x) = x^3 + 4x^2 + \sqrt{x} + 1$. 
Check your answer by differentiating.  Is your 
function $F(x)$ the \emph{only} antiderivative of $f$?  Why or why not?
\item[(2)] Find an antiderivative for $f(x) = e^{-4x} + e^{2x}$.  Check your
answer by differentiating.
\item[(3)] Find an antiderivative for $f(x) = \cos(x) + \sec^2(x)$.
\item[(4)]  Find an antiderivative for $\displaystyle f(x) = \frac{5}{\sqrt{1 - x^2}} + \frac{7}{1 + x^2}$.
\item[(5)]  Find an antiderivative for $\displaystyle f(x) = \frac{1}{x + 5}$.
Check your answer by differentiating. What if it was $\displaystyle f(x) = \frac{1}{2x + 5}$?
\item[(6)]  Find an antiderivative of $f(x) = \sin(x)$ and use it 
to evaluate 
$$\int_0^{\pi} \sin(x)\ dx.$$
Interpret via areas.  (Note:  Your answer should be strictly positive.  Why?)

\end{enumerate}

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