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True or False Integration by parts is based on the Product Rule for derivatives.

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The integration by parts formula states that \(\int\! u\,dv\,{=}\) ______.

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\(\int xe^{2x} dx\)

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\(\int xe^{-3x} dx\)

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\(\int x\cos x~dx\)

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

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\(\int \sqrt{x}\ln x~dx\)

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\(\int x^{-2}\ln x~dx\)

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\(\int \cot^{-1}x~dx\)

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\(\int \sin^{-1}x~dx\)

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\(\int (\ln x)^{2} dx\)

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\(\int x(\ln x)^{2} dx\)

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\(\int x^{2} \sin x~dx\)

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\(\int x^{2}cosx~dx\)

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\(\int x\cos^{2}x~dx\)

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\(\int x\sin^{2}x~dx\)

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\(\int x\sinh x~dx\)

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\(\int x\cosh x~dx\)

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\(\int \cosh^{-1}x~dx\)

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\(\int \sinh^{-1}x~dx\)

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\(\int \sin(\ln x)~dx\)

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\(\int \cos (\ln x)~dx\)

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\(\int (\ln x)^{3} dx\)

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\(\int (\ln x)^{4} dx\)

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\(\int x^{2}(\ln x)^{2} dx\)

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\(\int x^{3}(\ln x)^{2} dx\)

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\(\int x^{2}\tan ^{-1}x~dx\)

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\(\int x\tan ^{-1}x~dx\)

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\(\int 7^{x} x~dx\)

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\(\int 2^{-x} x~dx\)

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

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\(\int_{0}^{1}x^{2} e^{-x}dx\)

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\(\int_{0}^{2}x^{2} e^{-3x} dx\)

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\(\int_{0}^{\pi/4}x\tan ^{2}x~dx\)

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\(\int_{1}^{9}\ln \sqrt{x} dx\)

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\(\int_{\pi/4}^{3\pi/4}x\csc ^{2} x~dx\)

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\(\int^{e}_1 (\ln x)^{2} dx\)

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\(\int_{0}^{\pi/4}x \sec^{2} x~dx\)

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\(f(x)=3\ln x\) and \(g(x)=x\ln x, x \ge 1\)

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\(f(x)=4x\ln x\) and \(g(x)=x^{2}\ln x, x \ge 1\)

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Area Under a Graph Find the area under the graph of \(y=e^{x}\sin x\) from \(0\) to \(\pi\).

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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 \(y=\cos x\) and the \(x\)-axis from \(x=0\) to \(x=\dfrac{\pi }{2}\) about the \(y\)-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 graph of \(y=\sin x\) and the \(x\)-axis from \(x=0\) to \(x=\dfrac{\pi }{2}\) about the \(y\)-axis.

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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 \(y=x \sqrt{\sin x}\) and the \(x\)-axis from \(x=0\) to \(x=\dfrac{\pi }{2}\) about the \(x\)-axis.

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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 \(y=\ln x\) and the \(x\)-axis from \(x=1\) to \(x=e\) about the \(x\)-axis.

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Area

1. Graph the functions \(f( x)=x^{3}e^{-3x}\) and \(g( x) =x^{2}e^{-3x}\) on the same set of coordinate axes.
2. Find the area enclosed by the graphs of \(f\) and \(g\).

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Damped Spring The displacement \(x\) of a damped spring at time \(t\), \(0 \le t \le 5\), is given by \(x=x( t)=3e^{-t}\cos (2t) +2e^{-t}\sin (2t)\).

1. Graph \(x=x( t)\).
2. Find the least positive number \(t\) that satisfies \(x(t) =0\).
3. Find the area under the graph of \(x=x(t)\) from \(t=0\) to the value of \(t\) found in (b).

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A function \(y=f( x)\) is continuous and differentiable on the interval \(( 2,6)\). If \(\int_{3}^{5}f( x)~dx=18\) and \(f( 3) =8\) and \(f( 5) =11,\) then find \(\int_{3}^{5}x f^{\prime}(x)~dx\).

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\(\int \sin\sqrt{x}dx\)

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\(\int e^{\sqrt{x}}dx\)

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\(\int \cos x \ln(\sin x)~dx\)

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\(\int e^{x}\ln (2+e^{x}) dx\)

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

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\(\int \cos x\tan ^{-1} (\sin x)~dx\)

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Find \(\int x^{3}e^{x^{2}} dx\). (Hint: Let \(u=x^{2},dv=xe^{x^{2}} dx\).)

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Find \(\int x^{n} \ln x~dx\); \(n \neq -1\), \(n\) real.

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Find \(\int xe^{x}\cos x~dx\).

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Find \(\int xe^{x}\sin x~dx\).

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\(\int x^{n}\sin ^{-1}x\,dx=\dfrac{x^{n+1}}{n+1}\sin^{-1}x-\dfrac{1}{n+1}\int \dfrac{x^{n+1}}{\sqrt{1-x^{2}}}\,dx\)

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\(\int \dfrac{dx}{(x^{2}+1)^{n+1}}=\left( 1-\dfrac{1}{2n}\right) \int \dfrac{dx}{(x^{2}+1)^{n}}+\dfrac{x}{2n(x^{2}+1)^{n}}\)

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\(\int \sin^{n}x~dx=-\dfrac{\sin ^{n-1}x\cos x}{n}+\dfrac{n-1}{n}\int \sin ^{n-2}x\,dx\)

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\(\int \sin^{n}x \cos^{m} x~dx = -\dfrac{\sin^{n-1}x \cos^{m+1}x}{n+m} + \dfrac{n-1}{n+m}\int \sin^{n-2}x \cos^{m} xdx\)

where \(m \neq -n\), \(m \neq -1\)

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1. Find \(\int x^{2}e^{5x}\,dx\).
2. Using integration by parts, derive a reduction formula for \(\int x^{n}e^{kx}\,dx\), where \(k \neq 0\) and \(n \ge 2\) is an integer, in which the resulting integrand involves \(x^{n-1}\).

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1. Assuming there is a function \(p\) for which \(\int x^{3}e^{x}dx\) \(=p( x) e^{x}\), show that \(p( x)+p^{\prime} (x) =x^{3}\).
2. Use integration by parts to find a polynomial \(p\) of degree 3 for which \(\int x^{3}e^{x}dx=p( x) e^{x}+C.\)

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1. Use integration by parts with \(u=\sin x\) and \(dv=\cos x\,dx\) to find a function \(f\) for which \(\int \sin x\cos x~dx=f( x) +C_{1}\).
2. Use integration by parts with \(u=\cos x\) and \(dv=\sin x\,dx\) to find a function \(g\) for which \(\int \sin x\cos x~dx=g( x) +C_{2}\).
3. Use the trigonometric identity \(\sin (2x) =2\sin x\cos x\) and substitution 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 the formula \[ \int \ln (x+\sqrt{x^{2}+a^{2}})\,dx = x\ln (x+\sqrt{x^{2}+a^{2}}) -\sqrt{x^{2}+a^{2}}+C \]

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Derive the formula \[ \int e^{ax}\sin (bx)~dx = \dfrac{e^{ax}[a\sin (bx)-b\cos (bx)]}{a^{2}+b^{2}} +C, a \gt 0, b\gt 0 \]

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Suppose \(F(x)=\int_{0}^{x}t\,g^{\prime }(t)\,dt\) for all \(x \ge 0\). Show that \(F(x)=\) \(xg(x)-\int_{0}^{x}g(t)\,dt\).

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Use Wallis’ formulas, given below, to find each definite integral.

• \(\int_{0}^{\pi/2} \sin^{n}x~dx = \int_{0}^{\pi /2} \cos^{n}x~dx\) \(n \gt 1\) an integer \[=\left\{ \begin{array}{@{}l@{\qquad}l} \dfrac{(n-1)(n-3)\cdots (4)(2)}{n(n-2)\cdots (5)(3)(1)} & n \gt 1 \hbox{ is odd} \\ \dfrac{(n-1)(n-3)\cdots (5)(3)(1)}{n(n-2)\cdots (4)(2)}\left( \dfrac{\pi }{2} \right) & n\gt1\hbox{ is even} \end{array} \right.\]

1. \(\int_{0}^{\pi /2}\sin ^{6}x~dx\)
2. \(\int_{0}^{\pi /2}\sin ^{5}x~dx\)
3. \(\int_{0}^{\pi /2}\cos ^{8}x~dx\)
4. \(\int_{0}^{\pi /2}\cos ^{6}x~dx\)

Challenge Problems

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Derive Wallis’ formulas given in Problem 69. (Hint: Use the result of Problem 61.)

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1. If \(n\) is a positive integer, use integration by parts to show that there is a polynomial \(p\) of degree \(n\) for which \[ \int x^{n}e^{x}dx=p( x) e^{x}+C \]

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2. Show that \(p( x) +p^{\prime} ( x) =x^{n}\).
3. Show that \(p\) can be written in the form \[ p(x) = \sum \limits_{k=0}^{n}(-1)^{k} \dfrac{n!}{(n-k)!}x^{n-k} \]

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Show that for any positive integer \(n\), \[ \begin{eqnarray*} \hspace{-20pc}\int_{0}^{1}e^{x^{2}} dx \\ = e\cdot \!\left[ 1-\frac{2}{3}+\frac{4}{15}-\frac{8}{105}+\cdots +\frac{(-1)^{n}2^{n}}{(2n+1)(2n-1)\cdots 3\cdot 1}\right] \\+ (-1)^{n+1} \cdot \frac{2^{n+1}}{(2n+1)(2n-1)\cdots 3\cdot 1} \int_{0}^{1}x^{2n+2}e^{x^{2}}dx \end{eqnarray*} \]

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Use integration by parts to show that if \(f\) is a polynomial of degree \(n \ge 1\), then \(\int f( x) e^{x}dx=g( x) e^{x}+C\) for some polynomial \(g( x)\) of degree \(n\).

Question

Start with the identity \(f(b)-f(a)=\int_{a}^{b}f^{\prime}(t)\,dt\) and derive the following generalizations of the Mean Value Theorem for Integrals:

• (a) \(f(b)-f(a)=f^{\prime} (a)(b-a)- \int_{a}^{b}f^{\prime \prime}(t)(t-b)\,dt\)
• (b) \( f(b)-f(a)=f^{\prime} (a)(b-a)+\dfrac{f^{\prime\prime} (a)}{2}(b-a)^{2} + \int_{a}^{b}\dfrac{f^{\prime \prime \prime} (t)}{2}(t-b)^{2} dt \)

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If \(y=f(x)\) has the inverse function given by \(x=f^{-1}(y)\), show that \[ \begin{equation*} \int_{a}^{b}f(x)\,dx+\int_{f(a)}^{f(b)}f^{-1}(y)\,dy=bf(b)-af(a) \end{equation*} \]

Question

1. When integration by parts is used to find \(\int e^{x}\cosh x~dx\), what happens? Explain.
2. Find \(\int e^{x}\cosh x~dx\) without using integration by parts.