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%  Inverse Trigonometric Functions                 %
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\problemgrouptitle{Inverse Trigonometric Functions}

\begin{problemblock}[2]{Find the exact value of the following expressions.}

\item $\displaystyle \sin ^{-1} \left( \frac{\sqrt 2}{2} \right)$    
\answer{
$\displaystyle \frac \pi 4$
}

\item $\displaystyle \tan ^{-1} \left( \sin \frac \pi 2 \right)$    
\answer{
$\displaystyle \frac \pi 4$
}

\item $\displaystyle \sin  \left(2 \cos ^{-1} \frac{\sqrt 3}{2} \right)$    
\answer{
$\displaystyle \frac {\sqrt {3} }2$
}

\item $\displaystyle \tan \left(\cos ^{-1} x \right)$    
\answer{
$\displaystyle \frac {\sqrt{1-x^2}}{x}$
}

\end{problemblock}

\begin{problemblock}[2]{Differentiate the given functions.}

\item $\displaystyle f(x)=\sin ^{-1} \left(e^x+2 \right)$
\answer{
$\displaystyle f'(x)=\frac{e^x}{\sqrt{1-\left(e^x+2 \right)^2}}$
}

\item $\displaystyle y= \cos ^{-1} \left(\ln x+5 \right)$
\answer{
$\displaystyle y'=- \frac 1{x \sqrt{1-(\ln x+5)^2}}$
}

\item $\displaystyle g(x)=\tan ^{-1} \left(\sin x\right)$
\answer{
$\displaystyle g'(x)=\frac{\cos x}{1+\sin ^2 x}$
}

\item $\displaystyle y=\left (\sin ^{-1} x \right)^{\ln x}$
\answer{
$\displaystyle y'=\left (\sin ^{-1} x \right)^{\ln x} \left [ \frac {\ln \left( \sin^{-1} x\right)}{x}+\frac {\ln x} {\sin^{-1} x \hspace {.1in}\sqrt{1-x^2}}\right]$
}

\item $\displaystyle \tan ^{-1} y +e^y+xy=5$
\answer{
$\displaystyle y'=-\frac {y \left(1+y^2 \right)}  {1+\left(x+e^y\right)\left(1+y^2 \right)}$
}

\item $\displaystyle \sin ^{-1} x +\sin^{-1} y=x^2+y^2$
\answer{
$\displaystyle y'=\frac {2x-\frac 1{\sqrt{1-x^2}}} {\frac 1{\sqrt{1-y^2}}-2y}$
}

\item $\displaystyle f(x)=\sec^{-1} (2x)$
\answer{
$\displaystyle f'(x)=\frac 1{x \sqrt{4x^2-1}}$
}

\end{problemblock}

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

\item $\displaystyle \int \frac {e^x}{3+2 e^{2x}} \,dx$
\answer{
$\displaystyle \frac 1{\sqrt{6}} \tan^{-1} \left(\sqrt{\frac 23} e^x \right)+C$
}

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

\item $\displaystyle \int \frac {3x}{\sqrt{1-x^4}} \,dx$
\answer{
$\displaystyle \frac 32  \sin^{-1} \left(x^2 \right)+C$
}

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

\item $\displaystyle \int \frac {\sin^{-1} x}{\sqrt{1-x^2}} \,dx$
\answer{
$\displaystyle \frac 12 \left( \sin^{-1} x \right)^2+C$
}

\item $\displaystyle \int \frac {\sec^2\! x}{\sqrt {1-\tan^2 x}} \,dx$
\answer{
$\displaystyle \sin^{-1} \left( \tan x \right)+C$
}

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

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

\item $\displaystyle \int_0^{\sqrt {2}/2} \frac {2x}{\sqrt {1-x^4}} \,dx$
\answer{
$\displaystyle  \frac {\pi}{6}$
}

\end{problemblock}

\textbook{\addtoanswerkey{\clearpage}}

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