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\problemgrouptitle{Integration by Parts}

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

\item $\displaystyle \int \,x^2 \, e^{-5x}\, dx$                  
\answer{
$\displaystyle -\frac 15 x^2 e^{-5x}-\frac 2{25}xe^{-5x}-\frac 2{125} e^{-5x}+C$
}

\item  $\displaystyle  \int {x^2\, \cos^{-1}x\,dx}$		
\answer{
$\displaystyle \frac 13 x^3 \cos^{-1} x-\frac 13 \left(1-x^2 \right)^{1/2}+\frac 19 \left(1-x^2\right)^{3/2}+C$
}

\item $\displaystyle\int{ \sqrt t\, e^{2\sqrt t} \, dt}$
\answer{
$\displaystyle  te^{2\sqrt t}-\sqrt t\, e^{2\sqrt t}+\frac 12 \, e^{2\sqrt t}+C$
}

\item $\displaystyle\int{x^{10} \ln x \,dx}$
\answer{
$\displaystyle \frac 1{11} x^{11} \ln x-\frac 1{121} x^{11}+C$
}

\item $\displaystyle\int{x^5\, \sin{x^3}\,dx}$
\answer{
$\displaystyle -\frac 1{3} x^{3} \cos \left(x^3\right)+\frac 13 \sin \left(x^3\right)+C$
}

\item $\displaystyle\int{e^{2x}\, \sin 4x \,dx}$
\answer{
$\displaystyle -\frac 15 e^{2x}\, \cos 4x+\frac 1{10}\, e^{2x}\,\sin 4x+C$
}

\item $\displaystyle\int_{\pi/4}^{\pi/3}{x\, \sec^2\! x\, dx}$
\answer{
$\displaystyle \frac{\sqrt 3\,\pi}{3}-\frac {\pi}4-\frac 12 \ln 2$
}

\item $\displaystyle\int{\sin (3\ln x) \,dx}$
\answer{
$\displaystyle \frac 1{10} \, x\, \sin (3\ln x)-\frac 3{10} \, x\,\cos (3\ln x)+C$
}

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

\item $\displaystyle\int_0^{\sqrt 3}\tan^{-1}\!x\,dx$
\answer{
$\displaystyle \frac{\sqrt 3\,\pi}{3}-\ln 2$
}

\end{problemblock}

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