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\(\int \frac{dx}{x^{2}+4x+20}\)

Question

\(\int \frac{y+1}{y^{2}+y+1}\,dy\)

Question

\(\int \sec ^{3}\phi \tan \phi \,d\phi\)

Question

\(\int \cot^{2}\theta \,\csc \theta\, d\theta \)

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\(\int \sin ^{3}\phi \,d\phi\)

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\(\int \frac{x^{2}}{\sqrt{4-x^{2}}}dx\)

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\(\int \frac{dx}{\sqrt{(x+2) ^{2}-1}}\)

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\(\int_{0}^{\pi /4}x\sin (2x) \,dx\)

Question

\(\int v\csc ^{2}v\,dv\)

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

Question

\(\int (4-x^{2})^{3/2}\,dx\)

Question

\(\int \frac{3x^{2}+1}{x^{3}+2x^{2}-3x}dx\)

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\(\int \frac{e^{2t}\,dt}{e^{t}-2}\)

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\(\int \frac{dy}{5+4y+4y^{2}}\)

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\(\int \frac{x\,dx}{x^{4}-16}\)

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

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\(\int \frac{y^{2}\,dy}{(y+1)^{3}}\)

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\(\int \frac{dx}{x^{2}\sqrt{x^{2}+25}}\)

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

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\(\int \frac{dx}{\sqrt{16+4x-2x^{2}}}\)

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\(\int \ln (1-y)\,dy\)

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\(\int \frac{x^{3}-2x-1}{(x^{2}+1) ^{2}}dx\)

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\(\int \frac{3x^{2}+2}{x^{3}-x^{2}}dx\)

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\(\int \frac{dy}{\sqrt{2+3y^{2}}}\)

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\(\int x^{2}\sin ^{-1}x\,dx\)

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\(\int \sqrt{16+9x^{2}}\,dx\)

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\(\int \frac{dx}{x^{2}+2x}\)

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\(\int \sin ^{4}y\cos ^{4}y\,dy\)

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\(\int \frac{w-2\,}{1-w^{2}}dw\)

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\(\int \frac{x}{\sqrt{x^{2}-4}}dx\)

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\(\int \frac{1}{\sqrt{x}}\cos ^{2}\sqrt{x}dx\)

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

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

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\(\int_{0}^{1}\frac{x^{2}}{\sqrt{4-x^{2}}}dx\)

Question

\(\int_{0}^{\sqrt{3}}\frac{x\,dx}{\sqrt{1+x^{2}}}\)

Question

1. Find \(\int \dfrac{\cos ^{2}(2x)~dx}{\sin ^{3}(2x) }\) using a Table of Integrals.
2. Find \(\int \dfrac{\cos ^{2}(2x)~dx}{\sin ^{3}(2x) }\) using a computer algebra system (CAS).
3. Verify the results from (a) and (b) are equivalent.

Question

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

Question

\[ \begin{align*} \int x^{n} (ax+b) ^{1/2}dx&=\dfrac{2x^{n}(ax+b) ^{3/2}}{(2n+3) a} \\ &\quad-\dfrac{2bn}{(2n+3) a} \int x^{n-1}(ax+b) ^{1/2}dx \end{align*} \]

Question

\(\int_{1}^{\infty }\dfrac{e^{-\sqrt{x}}}{\sqrt{x}}dx\)

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\(\int_{0}^{1}\dfrac{\sin \sqrt{x}}{\sqrt{x}}dx\)

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\(\int_{0}^{1}\dfrac{x\,dx}{\sqrt{1-x^{2}}}\)

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\(\int_{-\infty }^{0}xe^{x}\,dx\)

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Show that \(\int_{0}^{\pi /2}\dfrac{\sin x}{\cos x}\,dx\) diverges.

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Show that \(\int_{1}^{\infty }\dfrac{\sqrt{1+x^{1/8}}}{x^{3/4}}\,dx\) diverges.

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\(\int_{1}^{\infty }\dfrac{1+e^{-x}}{x}dx\)

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\(\int_{0}^{\infty }\dfrac{x}{(1+x) ^{3}}dx\)

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If \(\int x^{2}\cos x\,dx=f(x)-\int 2x\sin x\,dx\), find \(f\).

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Area and Volume

1. Find the area \(A\) of the region \(R\) bounded by the graphs of \(y=\ln x,\) the \(x\)-axis, and the line \(x=e.\)
2. Find the volume of the solid of revolution generated by revolving \(R\) about the \(x\)-axis.
3. Find the volume of the solid of revolution generated by revolving \(R\) about the \(y\)-axis.

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Arc Length Approximate the arc length of \(y=\cos x\) from \(x=0\) to \(x=\dfrac{\pi }{2}\).

1. using the Trapezoidal Rule with \(n=3\).
2. using Simpson’s Rule with \(n=4.\)

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Distance The velocity \(v\) (in meters per second) of a particle at time \(t\) is given in the table. Use the Trapezoidal Rule to approximate the distance traveled from \(t=1\) to \(t=4.\)

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Area Find the area, if it exists, of the region bounded by the graphs of \(y=x^{-2/3},\) \(y=0,\) \(x=0\), and \(x=1.\)

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Volume Find the volume, if it exists, of the solid of revolution generated when the region bounded by the graphs of \(y=x^{-2/3}\), \( y=0,\) \(x=0\), and \(x=1\) is revolved about the \(x\)-axis.