Determine the integral $\displaystyle \int \cos \theta \cos^5 (\sin \theta) d \theta$
Let $u = \sin \theta$, then $du = \cos \theta d \theta$. Thus,
$
\begin{equation}
\begin{aligned}
\int \cos \theta \cos^5 (\sin \theta) d \theta =& \int \cos^5 u du
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\\
\int \cos^5 u du =& \int \cos^4 u \cos u du
\\\\
\int \cos^5 u du =& \int (\cos^2 u)^2 \cos u du \qquad \text{Apply Pythagorean Idendity } \cos^2 u + \sin^2 u = 1
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\int \cos^5 u du =& \int (1 - \sin^2 u)^2 \cos u du
\end{aligned}
\end{equation}
$
Let $v = \sin u$, then $dv = \cos u du$. Thus,
$
\begin{equation}
\begin{aligned}
\int (1 - \sin^2 u)^2 \cos u du =& \int (1 - v^2)^2 dv &&
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\int (1 - \sin^2 u)^2 \cos u du =& \int (1 - 2v^2 + v^4) dv &&
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\int (1 - \sin^2 u)^2 \cos u du =& v - \frac{2v^{2 + 1}}{2 + 1} + \frac{v^{4 + 1}}{4 + 1} + c &&
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\int (1 - \sin^2 u)^2 \cos u du =& v - \frac{2v^3}{3} + \frac{v^5}{5} + c
&& \text{Substitute value of } v
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\int (1 - \sin^2 u)^2 \cos u du =& \sin u - \frac{2 (\sin )^3}{3} + \frac{(\sin u)^5}{5} + c
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\int (1 - \sin^2 u)^2 \cos u du =& \sin u - \frac{2 \sin^3 u}{3} + \frac{\sin^5 u}{u} + c
&& \text{Substitute value of } u
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\int (1 - \sin^2 u)^2 \cos u du =& \sin (\sin \theta) - \frac{2 \sin^3 (\sin \theta)}{3} + \frac{\sin^5 (\sin \theta)}{5} + c
\end{aligned}
\end{equation}
$
@ 2nd term
$
\begin{equation}
\begin{aligned}
\frac{1}{8} \int^{\pi}_0 \cos 2t dt =& \frac{1}{8} \int^{2 \pi}_0 \cos u \cdot \frac{du}{2}
\\
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\frac{1}{8} \int^{\pi}_0 \cos 2t dt =& \frac{1}{16} \int^{2 \pi}_0 \cos u du
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\frac{1}{8} \int^{\pi}_0 \cos 2t dt =& \frac{1}{16} \left[ \sin u \right]^{2 \pi}_0
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\frac{1}{8} \int^{\pi}_0 \cos 2t dt =& \frac{1}{16} (\sin 2 \pi - \sin 0)
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\frac{1}{8} \int^{\pi}_0 \cos 2t dt =& \frac{1}{16} (0)
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\frac{1}{8} \int^{\pi}_0 \cos 2t dt =& 0
\end{aligned}
\end{equation}
$
@ 3rd term
$
\begin{equation}
\begin{aligned}
\frac{1}{8} \int^{\pi}_0 \cos^2 2t dt =& \frac{1}{8} \int^{2 \pi}_0 \cos^2 u \cdot \frac{du}{2}
\\
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\frac{1}{8} \int^{\pi}_0 \cos^2 2t dt =& \frac{1}{16} \int^{2 \pi}_0 \cos^2 u du
\qquad \text{Apply half-angle formula } \cos 2 u = 2 \cos^2 u - 1
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\frac{1}{8} \int^{\pi}_0 \cos^2 2t dt =& \frac{1}{16} \int^{2 \pi}_0 \left(\frac{\cos 2 u + 1}{2} \right) du
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\frac{1}{8} \int^{\pi}_0 \cos^2 2t dt =& \frac{1}{32} \int^{2 \pi}_0 (\cos 2u + 1) du
\end{aligned}
\end{equation}
$
Let $v = 2u$, then $dv = 2 du$, so $\displaystyle du = \frac{dv}{2}$. When $u = 0, v = 0$ and when $u = 2 \pi, v = 4 \pi$
$
\begin{equation}
\begin{aligned}
\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{1}{32} \int^{4 \pi}_0 (\cos v + 1) \cdot \frac{dv}{2}
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\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{1}{64} \int^{4 \pi}_0 (\cos v + 1) dv
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\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{1 }{64} \left[ \sin v + v \right]^{4 \pi}_0
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\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{1}{64} (\sin 4 \pi + 4 \pi - \sin 0 - 0)
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\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{1}{64} (0 + 4 \pi - 0 - 0)
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\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{4 \pi}{64}
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\frac{1}{32} \int^{32}_0 (\cos 2u + 1) du =& \frac{\pi}{16}
\end{aligned}
\end{equation}
$
@ 4th term
$
\begin{equation}
\begin{aligned}
\frac{1}{8} \int^{\pi}_0 \cos^3 2t dt =& \frac{1}{8} \int^{2 \pi}_0 \cos^3 u \cdot \frac{du}{2}
\\
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\frac{1}{8} \int^{\pi}_0 \cos^3 2t dt =& \frac{1}{16} \int^{2 \pi}_0 \cos^3 u du
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\frac{1}{8} \int^{\pi}_0 \cos^3 2t dt =& \frac{1}{16} \int^{2 \pi}_0 (\cos^2 u)(\cos u) du
\qquad \text{Apply Trigonometric Identities } \cos^2 u + \sin^2 u = 1
\\
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\frac{1}{8} \int^{\pi}_0 \cos^3 2t dt =& \frac{1}{16} \int^{2 \pi}_0 (1 - \sin^2 u)(\cos u) du
\end{aligned}
\end{equation}
$
Let $v = \sin u$, then $dv = \cos u du$. When $u = 0, v = 0$ and when $u = 2 \pi, v = 0$. Therefore,
$
\begin{equation}
\begin{aligned}
\frac{1}{16} \int^{2 \pi}_0 (1 - \sin^2 u)(\cos u du) =& \frac{1}{16} \int^0_0 (1 - v^2) dv
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\frac{1}{16} \int^{2 \pi}_0 (1 - \sin^2 u)(\cos u du) =& \frac{1}{16} \left[ v - \frac{v^3}{3} \right]^0_0
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\frac{1}{16} \int^{2 \pi}_0 (1 - \sin^2 u)(\cos u du) =& \frac{1}{16} (0)
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\frac{1}{16} \int^{2 \pi}_0 (1 - \sin^2 u)(\cos u du) =& 0
\end{aligned}
\end{equation}
$
Combine the results of integration term by term
$
\begin{equation}
\begin{aligned}
\int^{\pi}_0 \sin^2 t \cos^4 t dt =& \frac{\pi}{8} + 0 - \frac{\pi}{16} - 0
\\
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\int^{\pi}_0 \sin^2 t \cos^4 t dt =& \frac{2 \pi + 0 - \pi - 0}{16}
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\int^{\pi}_0 \sin^2 t \cos^4 t dt =& \frac{\pi}{16}
\end{aligned}
\end{equation}
$
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