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Section 11
Integration by Use of Tables
Make the given changes in the indicated examples of this section, and then state which formula from Appendix D would be used to complete the integration.In Example $1,$ change the denominator to $(2+3 x)^{2}$.
Make the given changes in the indicated examples of this section, and then state which formula from Appendix D would be used to complete the integration.In Example $2,$ in the numerator, change $-$ to $+$.
Identify $u, d u,$ and the formula from Appendix $D$ that would be used to complete the integration. Do not integrate.$$\int \frac{4 d y}{3 y \sqrt{1+2 y}}$$
Identify $u, d u,$ and the formula from Appendix $D$ that would be used to complete the integration. Do not integrate.$$\int \frac{x d x}{\sqrt{x^{4}-16}}$$
Identify $u, d u,$ and the formula from Appendix $D$ that would be used to complete the integration. Do not integrate.$$\int \frac{x d x}{\left(4-x^{4}\right)^{3 / 2}}$$
Identify $u, d u,$ and the formula from Appendix $D$ that would be used to complete the integration. Do not integrate.$$\int x^{5} \ln x^{3} d x$$
Identify $u, d u,$ and the formula from Appendix $D$ that would be used to complete the integration. Do not integrate.$$\int x \cos ^{2}\left(x^{2}\right) d x$$
Identify $u, d u,$ and the formula from Appendix $D$ that would be used to complete the integration. Do not integrate.$$\int \frac{d s}{s\left(s^{4}-1\right)^{3 / 2}}$$
Integrate each function by using the table in Appendix $D$.$$\int \frac{3 x d x}{2+5 x}$$
Integrate each function by using the table in Appendix $D$.$$\int \frac{4 x d x}{1+2 x+x^{2}}$$
Integrate each function by using the table in Appendix $D$.$$\int_{2}^{7} 4 x \sqrt{2+x} d x$$
Integrate each function by using the table in Appendix $D$.$$\int \frac{d x}{x^{2}-4}$$
Integrate each function by using the table in Appendix $D$.$$\int \frac{8 d y}{\left(y^{2}+4\right)^{3 / 2}}$$
Integrate each function by using the table in Appendix $D$.$$\int_{0}^{\pi / 3} 3 \sin ^{3} x d x$$
Integrate each function by using the table in Appendix $D$.$$\int \sin 2 x \sin 3 x d x$$
Integrate each function by using the table in Appendix $D$.$$\int 6 \sin ^{-1} 3 x d x$$
Integrate each function by using the table in Appendix $D$.$$\int \frac{\sqrt{4 x^{2}-9}}{x} d x$$
Integrate each function by using the table in Appendix $D$.$$\int \frac{\left(9 x^{2}+16\right)^{3 / 2}}{x} d x$$
Integrate each function by using the table in Appendix $D$.$$\int \cos ^{5} 4 x d x$$
Integrate each function by using the table in Appendix $D$.$$\int \tan ^{-1} 2 x d x$$
Integrate each function by using the table in Appendix $D$.$$\int 6 r \tan ^{-1} r^{2} d r$$
Integrate each function by using the table in Appendix $D$.$$\int 5 x e^{4 x} d x$$
Integrate each function by using the table in Appendix $D$.$$\int_{1}^{2}\left(4-x^{2}\right)^{3 / 2} d x$$
Integrate each function by using the table in Appendix $D$.$$\int \frac{3 d x}{9-16 x^{2}}$$
Integrate each function by using the table in Appendix $D$.$$\int \frac{d x}{2 x \sqrt{x^{2}+\frac{1}{4}}}$$
Integrate each function by using the table in Appendix $D$.$$\int \frac{\sqrt{9+x^{2}}}{x} d x$$
Integrate each function by using the table in Appendix $D$.$$\int \frac{8 d x}{x \sqrt{1-4 x^{2}}}$$
Integrate each function by using the table in Appendix $D$.$$\int \sqrt{5-16 x^{2}} d x$$
Integrate each function by using the table in Appendix $D$.$$\int_{0}^{\pi / 12} \sin \theta \cos 5 \theta d \theta$$
Integrate each function by using the table in Appendix $D$.$$\int_{0}^{2} x^{2} e^{3 x} d x$$
Integrate each function by using the table in Appendix $D$.$$\int 6 x^{5} \cos x^{3} d x$$
Integrate each function by using the table in Appendix $D$.$$\int 15 \sin ^{3} t \cos ^{2} t d t$$
Integrate each function by using the table in Appendix $D$.$$\int \frac{2 x d x}{\left(1-x^{4}\right)^{3 / 2}}$$
Integrate each function by using the table in Appendix $D$.$$\int \frac{d x}{x-4 x^{2}}$$
Integrate each function by using the table in Appendix $D$.$$\int_{1}^{3} \frac{\sqrt{3+5 x^{2}} d x}{x}$$
Integrate each function by using the table in Appendix $D$.$$\int_{1 / 2}^{1} \frac{\sqrt{9-4 x^{2}}}{x} d x$$
Integrate each function by using the table in Appendix $D$.$$\int x^{3} \ln x^{2} d x$$
Integrate each function by using the table in Appendix $D$.$$\int \frac{1.2 u d u}{u^{2} \sqrt{u^{4}-9}}$$
Integrate each function by using the table in Appendix $D$.$$\int \frac{9 x^{2} d x}{\left(x^{6}-1\right)^{3 / 2}}$$
Integrate each function by using the table in Appendix $D$.$$\int x^{7} \sqrt{x^{4}+4} d x$$
Integrate each function by using the table in Appendix $D$.$$\int t^{2}\left(t^{6}+1\right)^{3 / 2} d t$$
Integrate each function by using the table in Appendix $D$.$$\int \frac{\sqrt{3+4 x^{2}} d x}{x}$$
Integrate each function by using the table in Appendix $D$.$$\int \sin ^{3} 4 x \cos ^{3} 4 x d x$$
Integrate each function by using the table in Appendix $D$.$$\int 6 \cot ^{4} 2 d x$$
Integrate each function by using the table in Appendix $D$.A good representation of the cables between towers of the $2280-\mathrm{m}$ section of the Golden Gate Bridge is $y=0.000370 x^{2}$ for $-1140 \leq x \leq 1140,$ where $x$ and $y$ are in meters. Find the length of the cables (see Exercise 35 of Section 26.6 ).
Integrate each function by using the table in Appendix $D$.The design of a rotor can be represented as the volume generated by rotating the area bounded by $y=3.00 \ln x, x=3.00,$ and the $x$ -axis. Find its radius of gyration about the $y$ -axis.
Integrate each function by using the table in Appendix $D$.Find the area of an ellipse with a major axis $2 a$ and a minor axis $2 b$
Integrate each function by using the table in Appendix $D$.The voltage across a $5.0-\mu$ F capacitor in an electric circuit is zero. What is the voltage after $5.00 \mu$ s if a current $i$ (in $m A$ ) as a function of the time $t$ (in s) given by $i=\tan ^{-1} 2 t$ charges the capacitor?
Integrate each function by using the table in Appendix $D$.Find the force (in $1 \mathrm{b}$ ) on the region bounded by $x=1 / \sqrt{1+y}$ $y=0, y=3,$ and the $y$ -axis, if the surface of the water is at the upper edge of the area.
Integrate each function by using the table in Appendix $D$.If $6.00 \mathrm{g}$ of a chemical are placed in water, the time $t$ (in min) it takes to dissolve half of the chemical is given by $t=560 \int_{3}^{6} \frac{d x}{x(x+4)}$ where $x$ is the amount of undissolved chemical at any time. Evaluate $t$.
Integrate each function by using the table in Appendix $D$.The dome of a sports arena is the surface generated by revolving $y=20.0 \cos 0.0196 x(0 \leq x \leq 80.0 \mathrm{m})$ about the $y$ -axis. Find the volume within the dome.
Integrate each function by using the table in Appendix $D$.If an electric charge $Q$ is distributed along a wire of length $2 a$ the force $F$ exerted on an electric charge $q$ placed at point $P$ is $F=k q Q \int \frac{b d x}{\left(b^{2}+x^{2}\right)^{3 / 2}} .$ Integrate to find $F$ as a function of $x$.