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\problemgrouptitle{Integration Using All Methods}

\begin{problemblock}[2]{Evaluate the given integrals using any method.}

\item $\displaystyle \int \left(x^3+\sqrt{x}-\frac{1}{x^2}+5\right)\,dx$
\answer{$\displaystyle \frac{1}{4}x^4+\frac{2}{3}x^{\frac{3}{2}}+\frac{1}{x}+5x+C$}

\item $\displaystyle \int \left(x^3+2\right)^2\,dx$
\answer{$\displaystyle \frac{1}{7}x^7+x^4+4x+C$}

\item $\displaystyle \int \frac{\left(2x+\sqrt{x}\right)^2}{\sqrt{x}}\,dx$
\answer{$\displaystyle \frac{8}{5}x^{\frac{5}{2}}+2x^2+\frac{2}{3}x^{\frac{3}{2}}+C$}

\item $\displaystyle \int \left(\cos\theta+\sec^2\theta\right)\,d\theta$
\answer{$\displaystyle \sin\theta+\tan\theta+C$}

\item $\displaystyle \int 3x^2\left(x^3+4\right)^5\,dx$
\answer{$\displaystyle \frac{1}{6}\left(x^3+4\right)^6+C$}

\item $\displaystyle \int \cos\theta\left(\sin\theta+3\right)^{10}\,d\theta$
\answer{$\displaystyle \frac{1}{11}\left(\sin\theta+3\right)^{11}+C$}

\item $\displaystyle \int \frac{\left(\sqrt{t}+7\right)^{\frac{4}{3}}}{\sqrt{t}}\,dt$
\answer{$\displaystyle \frac{6}{7}\left(\sqrt{t}+7\right)^{\frac{7}{3}}+C$}

\item $\displaystyle \int_{-1}^2~\left(2x+3\right)^4\,dx$
\answer{$\displaystyle \frac{16806}{10}$}

\item $\displaystyle \int_0^1~\frac{x^2}{\left(x^3+2\right)^3}\,dx$
\answer{$\displaystyle \frac{5}{216}$}

\item $\displaystyle \int_0^{\frac{\pi }{4}}\sin\left(2\theta\right)\,d\theta$
\answer{$\displaystyle \frac{1}{2}$}

\item $\displaystyle \int_{-\frac{3}{2}}^{\frac{3}{2}}\sin(\tan x)\,dx$
\answer{$0$\ \  Note that since sine and tangent are odd, the integrand itself is odd.  Since the limits are~$\pm a$, the integral vanishes.}

\end{problemblock}
