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\problemgrouptitle{Integration Examples}
  
\begin{problemblock}[2]{Evaluate the given integral.}

\addtoanswerkey{\begin{multicols}{2}}
  
\item $\displaystyle \int \left(x^3+\frac 1{\sqrt x} +\frac 5{x^3}\right)\, dx$
\answer{
$\displaystyle \frac 14 x^4+2\sqrt x-\frac 5{2x^2}+C$
}

\item $\displaystyle \int \frac {x^3+\sqrt x+1}{\sqrt x}\, dx$
\answer{
$\displaystyle \frac 27 x^{7/2}+x+2 x^{1/2}+C$
}

\item $\displaystyle \int \frac {\tan\!\sqrt x \,\,\sec\!\sqrt x}{\sqrt x}\, dx$
\answer{
$\displaystyle 2\sec \sqrt x+C$
}

\item $\displaystyle \int_0^1 \left ( x^2-x^3 \right)\, dx$
\answer{
$\displaystyle \frac {1}{12}$
}

\item $\displaystyle \int_0^{\pi/2} \cos x \left ( 1+\sin x \right)^3 \, dx$
\answer{
$\displaystyle \frac {15}{4}$
}

\item $\displaystyle \int x^2 \sin \left(x^3+2\right)\, dx$
\answer{
$\displaystyle -\frac 13 \cos \left(x^3+2\right)+C$
}

\item $\displaystyle \int \sqrt{2x+5}\, dx$                     
\answer{
$\displaystyle \frac 13 \left(2x+5\right)^{3/2}+C$
}

\item $\displaystyle \int _0^3 x\left(x^2+16\right)^{3/2}\, dx$
\answer{
$\displaystyle \frac {2101}{5}$
}

\item $\displaystyle \int \frac{\sin(3x)}{\left(2+\cos(3x)\right)^2}\, dx$
\answer{
$\displaystyle \frac 13\,\frac 1{2+\cos3x}+C$
}

\item $\displaystyle \int _5^{12} \frac 1{(x-4)^{1/3}}\, dx$
\answer{
$\displaystyle \frac {9}{2}$
}

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

\item $\displaystyle \int x \left(x+4\right)^5\, dx$ 
\answer{
$\displaystyle \frac 17 \left(x+4\right)^7-\frac 23 \left(x+4\right)^6+C$
}

\addtoanswerkey{\end{multicols}}

\end{problemblock}

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