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\problemgrouptitle{General Strategies for Integration}

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

\item $\displaystyle\int\,\frac { e^{\sqrt x}}{\sqrt x}\, dx$
\answer{
$2\, e^{\sqrt x}+C$
}

\item $\displaystyle\int \sin \sqrt{x+5}\, dx$
\answer{
$\displaystyle -2 \sqrt{x+5} \cos \sqrt{x+5}+2\sin \sqrt{x+5}+C$
}

\item $\displaystyle\int \frac {e^{2x}}{3+e^x}\, dx$
\answer{
$\displaystyle 3+e^x-3\ln |3+e^x |+C$
}

\item $\displaystyle\int \frac {e^{3x}}{5+e^{6x}}\, dx$
\answer{
$\displaystyle \frac 1{3\sqrt5}\, \tan^{-1}\!\left( \frac {e^{3x}}{\sqrt 5} \right)+C$
}

\item $\displaystyle\int_1^4 \frac {1}{3+\sqrt x}\, dx$
\answer{
$\displaystyle 2-6\ln 5+6\ln 4$
}

\item $\displaystyle\int \frac {\sin 2x}{ \sin^2 x-\sin x-6}\, dx$
\answer{
$\displaystyle \frac 65 \ln |\sin x-3 |+\frac 45 \ln |\sin x+2 |+C$
}

\item $\displaystyle\int \frac {x^3+5x}{\left( x^2+1\right)^2}\, dx$
\answer{
$\displaystyle \frac 12  \ln | x^2+1| -\frac {2}{x^2+1}+C$
}

\item $\displaystyle\int \frac {e^x}{ \sqrt{e^{2x}+4e^x+6}}\, dx$
\answer{
$\displaystyle \ln \left | \frac {\sqrt{e^{2x}+4e^x+6}}{\sqrt 2}+\frac {e^x+2}{\sqrt2} \right |+C$
}

\item $\displaystyle\int_0^1 \frac {x+2}{ x^2+2x+3}\, dx$
\answer{
$\displaystyle \frac {\ln 6}2+\frac 1{\sqrt 2} \tan^{-1}\!\sqrt 2- \frac 12 \ln 3-\frac 1{\sqrt 2}\tan^{-1}\left(\frac {1}{\sqrt 2}\right)$
}

\item $\displaystyle\int \frac {4x}{\left ( x^2+2x+9\right)^2}\, dx$
\answer{
$\displaystyle -\frac {2}{(x+1)^2+8}-\frac {\sqrt 8}{32}\tan^{-1}\!\left( \frac{x+1}{\sqrt 8}\right)-\frac 14 \frac {x+1}{(x+1)^2+8}+C$
}

\end{problemblock}

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