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\problemgrouptitle{Trigonometric Integrals}

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

\item $\displaystyle \int \,\cos^{10}x\, \sin^5 x\, dx$
\answer{
$\displaystyle -\frac 1{11}\cos ^{11} x+\frac 2{13}\cos^{13}x-\frac 1{15}\, \cos^{15}x+C$
}

\item $\displaystyle \int \,\sin^{4}x\, \cos^3 x\, dx$
\answer{
$\displaystyle \frac 1{5}\sin ^{5} x-\frac 1{7}\sin^{7}x+C$
}

\item $\displaystyle \int \,\sin^{4}x\, dx$
\answer{
$\displaystyle \frac 38\, x-\frac 1{4}\sin 2 x+\frac 1{32}\sin 4x+C$
}

\item $\displaystyle \int \,\tan^{4}x\,\sec^4 x\, dx$
\answer{
$\displaystyle  \frac 1{5}\tan^5 x+\frac 1{7}\tan^7 x+C$
}

\item $\displaystyle \int \,\tan^{5}x\,\sec^5 x\, dx$
\answer{
$\displaystyle  \frac 1{9}\sec^9 x-\frac 2{7}\sec^7 x+\frac 15\sec^5 x+C$
}

\item $\displaystyle \int \,\cot^{2}x\,\csc^4 x\, dx$
\answer{
$\displaystyle  -\frac 1{3}\cot^3 x-\frac 1{5}\cot^5 x+C$
}

\item $\displaystyle \int \,\tan^{4}x\, dx$
\answer{
$\displaystyle  \frac 1{3}\tan^3 x-\tan x+x+C$
}

\item $\displaystyle \int \,\sin 9x\,\cos 7 x\, dx$
\answer{
$\displaystyle -\frac 1{4}\cos 2x-\frac 1{32}\cos 16 x+C$
}

\item $\displaystyle \int \,\sin 3x\,\sin 5 x\, dx$
\answer{
$\displaystyle \frac 1{4}\sin 2x-\frac 1{16}\sin 8 x+C$
}

\item $\displaystyle \int \,\cos 4x\,\cos 5 x\, dx$
\answer{
$\displaystyle  \frac 1{2}\sin x+\frac 1{18}\sin 9 x+C$
}

\end{problemblock}

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