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%  Fundamental Theorem of Calculus Antiderivative Form %
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\problemgrouptitle{Fundamental Theorem of Calculus (Antiderivative Form)}

\begin{problemblock}[2]{Compute the following definite integrals using the Fundamental Theorem of Calculus.}

\item $\displaystyle\int_1^3\left(x^2+3\right)dx$
\answer{$\displaystyle\frac{44}{3}$}

\item $\displaystyle\int_4^1\sqrt{x}\,dx$
\answer{$\displaystyle-\frac{14}{3}$}

\item $\displaystyle\int_{-\frac{\pi}{4}}^\frac{\pi}{4}\sec^2\theta\d{\theta}$
\answer{$2$}

\item $\displaystyle\int_{-2}^{-1}\frac{4}{x^4}\,dx$
\answer{$\displaystyle\frac{7}{6}$}	

\item $\displaystyle\int_{-\frac{\pi}{2}}^\frac{\pi}{2}\left(\sin x+x\right)\d{x}$
\answer{$0$}

\item $\displaystyle\int_{-3}^2|x|\d{x}$
\answer{$\displaystyle\frac{13}{2}$}

\item $\displaystyle\int_{-1}^1\frac{1}{x^2}\,dx$
\answer{The Fundamental Theorem of Calculus is not applicable here because the integrand $\frac{1}{x^2}$ is discontinuous on $[-1,1]$.  The area under the curve (i.e. the integral) in fact diverges to $+\infty$\,.  To show that requires a consideration of improper integrals.}  

\end{problemblock}
