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True or False To find \(\int \cos ^{5}x\,dx,\) factor out \(\cos x\) and use the identity \(\cos ^{2}x=1-\sin ^{2}x.\)

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True or False To find \(\int \sin (2x) \cos (3x) \,dx,\) use a product-to-sum identity.

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\(\int \cos ^{5}x\,dx\)

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\(\int \sin ^{3}x\,dx\)

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\(\int \sin ^{6}x\,dx\)

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\(\int \cos ^{4}x\,dx\)

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\(\int \sin ^{2}\left( \pi x\right) \,dx\)

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\(\int \cos ^{4}(2x) \,dx\)

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\(\int_{0}^{\pi }\cos ^{5}x~dx\)

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\(\int_{-\pi /3}^{\pi /3}\sin ^{3}x\,dx\)

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\(\int \sin ^{3}x\cos ^{2}x\,dx\)

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\(\int \sin ^{4}x\cos ^{3}x\,dx\)

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\(\int \sin ^{2}x\cos ^{2}x\,dx\)

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\(\int \sin ^{4}x\cos ^{2}x\,dx\)

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\(\int \sin x\cos ^{1/3}x\,dx\)

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\(\int \cos ^{3}x\sin ^{1/2}x\,dx\)

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\(\int \sin ^{2}\left(\dfrac{x}{2}\right) \cos ^{3}\left( \dfrac{x}{2}\right) \,dx\)

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\(\int \sin ^{3}(4x) \cos ^{3}(4x) \,dx \)

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\(\int \tan ^{3}x\sec ^{2}x\,dx\)

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\(\int \tan x\sec ^{5}x\,dx\)

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\(\int \tan ^{2}x\sec ^{2}x\,dx\)

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\(\int \tan ^{5}x\sec ^{2}x\,dx\)

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\(\int \tan ^{2}x\sec ^{3}x\,dx\)

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\(\int \tan ^{4}x\,\sec x\,dx\)

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\(\int \cot ^{3}x\csc x\,dx\)

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\(\int \cot ^{3}x\csc ^{2}x\,dx\)

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\(\int \sin (3x) \cos x\,dx\)

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\(\int \sin x\cos (3x) \,dx\)

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\(\int \cos x\cos (3x) \,dx\)

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\(\int \cos (2x) \cos x\,dx \)

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\(\int \sin (2x) \sin (4x) \,dx\)

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\(\int \sin (3x) \sin x\, dx\)

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\(\int_{0}^{\pi /2}\sin (2x) \sin x\,dx\)

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\(\int_{0}^{\pi }\cos x\cos (4x) \,dx\)

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\(\int \sin ^{2}x\cos x\,dx\)

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\(\int \sin ^{3}x\cos x\,dx\)

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\(\int \dfrac{\sin x\,dx}{\cos ^{2}x}\)

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\(\int \dfrac{\cos x\,dx}{\sin ^{4}x}\)

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\(\int \cos ^{3}(3x) \,dx\)

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\(\int \sin ^{5}(3x) \,dx\)

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\(\int_{0}^{\pi }\sin ^{3}x\cos ^{5}x\,dx\)

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\(\int_{0}^{\pi /2}\sin ^{3}x\cos ^{3}x\,dx\)

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\(\int \tan ^{3}x\,dx\)

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\(\int \cot ^{5}x\,dx\)

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\(\int \dfrac{\sec ^{6}x}{\tan ^{3}x}\,dx\)

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\(\int \tan ^{1/2}x\sec ^{2}x\,dx\)

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\(\int \csc ^{2}x\cot ^{5}x\,dx\)

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\(\int \cot x\csc ^{2}x\,dx\)

487

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\(\int \cot(2x) \csc^{4}(2x)~dx\)

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\(\int \cot^{2}(2x) \csc^{3}(2x)dx\)

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\(\int_{0}^{\pi /4}\tan^{4}x\sec^{3}xdx \)

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\(\int_{0}^{\pi /4}\tan^{2}x\sec xdx\)

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\(\int \sin \left( \dfrac{x}{2}\right) \cos \left( \dfrac{3x}{2}\right)dx\)

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\(\int \cos (-x) \sin (4x)dx\)

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\(\int \sin \left( \dfrac{x}{2}\right) \sin \left( \dfrac{3x}{2}\right)dx\)

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\(\int \cos (\pi x) \cos ( 3\pi x)dx\)

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Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region bounded by the graph of \(y=\sin x\) and the \(x\)-axis from \(x=0\) to \(x=\pi \) about the \(x\)-axis. See the figure below.

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Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of \(y=\cos x\), \(y=\sin x\), and \(x=0\) from \(x=0\) to \(x=\dfrac{\pi }{4}\) about the \(x\)-axis.

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Average Value

1. Find the average value of \(f( x) =\sin x\cos ^{4}x\) over the interval \([ 0,\,\pi ] \).
2. Give a geometric interpretation to the average value.
3. Use graphing technology to graph \(f\) and the average value on the same screen.

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Rectilinear Motion The acceleration \(a\) of an object at time \(t\) is given by \(a(t) =\cos ^{2}t\sin t\) m\(/\)s\(^{2}\). At \(t=0\), the object is at the origin and its speed is 5m\(/\)s. Find its distance from the origin at any time \(t\).

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Area and Volume Let \(A\) be the area of the region in the first quadrant bounded by the graphs of \(y=\sec x\), \(y=2\sin x\), and the \(y\)-axis.

1. Find \(A\).
2. Find the volume of the solid of revolution generated by revolving the region about the \(x\)-axis.

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1. Use technology to graph the function \(f( x) =\sin^{n}x,\) \(0 \le x \le \pi\), for \(n=5\), \(n=10\), \(n=20\), and \(n=50\).
2. Find \(\int_{0}^{\pi }\sin ^{n}xdx\) correct to three decimal places for \(n=5\), \(n=10\), \(n=20\), and \(n=50\).
3. What does (a) suggest about the shape of the graph of \(f( x) =\sin ^{n}x\), \(0 \le x \le \pi\), as \(n\rightarrow \infty\).
4. Find \(\lim\limits_{n\rightarrow \infty }\int_{0}^{\pi }\sin ^{n}x~dx\).

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Find \(\int \sin ^{4}x\,dx\).

1. Using the methods of this section.
2. Using the reduction formula given in Problem 61 in Section 7.1.
3. Verify that both results are equivalent.
4. Use a CAS to find \(\int \sin ^{4}x~dx\).

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1. Use the substitution \(u=\sin x\) to find a function \(f\) for which \[ \int \sin x\cos x\,dx=f(x)+C_{1}. \]
2. Use the substitution \(u=\cos x\) to find a function \(g\) for which \[ \int \sin x\cos x\,dx=g(x)+C_{2}. \]
3. Use the trigonometric identity \(\sin (2x)=2\sin x\cos x\) to find a function \(h\) for which \[ \int \sin x\cos x\,dx=h(x)+C_{3}. \]
4. Compare the functions \(f\) and \(g\). Find a relationship between \(C_{1}\) and \(C_{2}\).
5. Compare the functions \(f\) and \(h\). Find a relationship between \(C_{1}\) and \(C_{3}\).

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Derive a formula for \(\int \sin (mx) \sin (nx) \,dx,\quad m ≠ n\).

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Derive a formula for \(\int \sin (mx) \cos (nx) \,dx,\quad m ≠ n\).

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Derive a formula for \(\int \cos (mx) \cos (nx) \,dx,\quad m ≠ n\).

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Use the substitution \(\sqrt{x}\,=\,\sin y\) to find \(\int_{0}^{1/2}\dfrac{\sqrt{x}}{\sqrt{1-x}}\,dx\). \(\bigg(\)Hint: sin\(^{2}y\,{=}\,\dfrac{1\,{-}\,\cos (2y)}{2}.\bigg)\)

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Use an appropriate substitution to show that \[ \int_{0}^{\pi /2}\sin ^{n}\theta \,d\theta =\,\int_{0}^{\pi /2}\cos ^{n}\theta \,d\theta . \]

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1. What is wrong with the following?
   
   
2. Find \(\int_{0}^{\pi }\cos ^{4}x~dx\).