Chapter 5

\(\frac{\ln \sqrt{x}}{x} d x=\) (A) \(\frac{\ln ^{2} \sqrt{x}}{\sqrt{x}}+C\) (B) \(\frac{1}{2} \ln |\ln x|+C\) (C) \(\frac{(\ln \sqrt{x})^{2}}{2}+C\) (D) \(\frac{1}{4} \ln ^{2} x+C\)

\(\int x^{3} \ln x d x=\) (A) \(x^{2}(3 \ln x+1)+C\) (B) \(\frac{x^{4}}{16}(4 \ln x-1)+C\) (C) \(\frac{x^{4}}{4}(\ln x-1)+C\) (D) \(3 x^{2}\left(\ln x-\frac{1}{2}\right)+C\)

\(\int \ln \eta d \eta=\) (A) \(\frac{1}{2} \ln ^{2} \eta+C\) (B) \(\eta(\ln \eta-1)+C\) (C) \(\frac{1}{2} \ln \eta^{2}+C\) (D) \(\eta \ln \eta+\eta+C\)

\(\int \ln x^{3} d x=\) (A) \(\frac{3}{2} \ln ^{2} x+C\) (B) \(3 x(\ln x-1)+C\) (C) \(3 \ln x(x-1)+C\) (D) \(\frac{3 x \ln ^{2} x}{2}+C\)

\(\int \frac{\ln y}{y^{2}} d y=\) (A) \(\frac{1}{y}(1-\ln y)+C\) (B) \(\frac{1}{2 y} \ln ^{2} y+C\) (C) \(-\frac{1}{y}(\ln y+1)+C\) (D) \(\frac{\ln y}{y}-\frac{1}{y}+C\)

\(\int \frac{d v}{v \ln v}=\) (A) \(\frac{1}{\ln v^{2}}+C\) (B) \(-\frac{1}{\ln ^{2} v+C}\) (C) \(-\ln |\ln v|+C\) (D) \(\ln |\ln v|+C\)

\(\int \frac{y-1}{y+1} d y=\) (A) \(y-2 \ln |y+1|+C\) (B) \(1-\frac{2}{y+1}+C\) (C) \(\ln \frac{|y|}{(y+1)^{2}}+C\) (D) \(1-2 \ln |y+1|+C\)

\(\int \frac{d x}{x^{2}+2 x+2}=\) (A) \(\ln \left(x^{2}+2 x+2\right)+C\) (B) \(\ln |x+1|+C\) (C) \(\arctan (x+1)+C\) (D) \(\frac{1}{\frac{1}{3} x^{3}+x^{2}+2 x}+C\)

\(\int \sqrt{x}(\sqrt{x}-1) d x=\) (A) \(2\left(x^{3 / 2}-x\right)+C\) (B) \(\frac{x^{2}}{2}-x+C\) (C) \(\frac{1}{2}(\sqrt{x}-1)^{2}+C\) (D) \(\frac{1}{2} x^{2}-\frac{2}{3} x^{3 / 2}+C\)

\(\int e^{\theta} \cos \theta d \theta=\) (A) \(e^{\theta}(\cos \theta-\sin \theta)+C\) (B) \(\frac{1}{2} e^{\theta}(\sin \theta+\cos \theta)+C\) (C) \(2 e^{\theta}(\sin \theta+\cos \theta)+C\) (D) \(\frac{1}{2} e^{\theta}(\sin \theta-\cos \theta)+C\)

\(\int \frac{(1-\ln t)^{2}}{t} d t=\) (A) \(\frac{1}{3}(1-\ln t)^{3}+C\) (B) \(\ln t-2 \ln ^{2} t+\ln ^{3} t+C\) (C) \(\ln t-\ln ^{2} t+\frac{\ln t^{3}}{3}+C\) (D) \(-\frac{(1-\ln t)^{3}}{3}+C\)

\(\int u \sec ^{2} u d u=\) (A) \(u \tan u+\ln |\cos u|+C\) (B) \(\frac{u^{2}}{2} \tan u+C\) (C) \(\frac{1}{2} \sec u \tan u+C\) (D) \(u \tan u-\ln |\sin u|+C\)

\(\int \frac{2 x+1}{4+x^{2}} d x=\) (A) \(\ln \left(x^{2}+4\right)+C\) (B) \(\ln \left(x^{2}+4\right)+\tan ^{-1} \frac{x}{2}+C\) (C) \(\ln \left(x^{2}+4\right)+\frac{1}{2} \tan ^{-1} \frac{x}{2}+C\) (D) \(\ln \left(x^{2}+4\right)+\frac{1}{4} \tan ^{-1} x+C\)

\(\int \frac{1-x}{\sqrt{1-x^{2}}} d x=\) (A) \(\sqrt{1-x^{2}}+C\) (B) \(\sin ^{-1} x+C\) (C) \(\sin ^{-1} x+\sqrt{1-x^{2}}+C\) (D) \(\sin ^{-1} x+\frac{1}{2} \ln \sqrt{1-x^{2}}+C\)

\(\int \frac{2 x-1}{\sqrt{4 x-4 x^{2}}} d x=\) (A) \(4 \ln \sqrt{4 x-4 x^{2}}+C\) (B) \(\sin ^{-1}(1-2 x)+C\) (C) \(\frac{1}{2} \sqrt{4 x-4 x^{2}}+C\) (D) \(-\frac{1}{2} \sqrt{4 x-4 x^{2}}+C\)

\(\int \frac{\cos \theta}{1+\sin ^{2} \theta} d \theta=\) (A) \(\sec \theta \tan \theta+C\) (B) \(\ln \left(1+\sin ^{2} \theta\right)+C\) (C) \(\tan ^{-1}(\sin \theta)+C\) (D) \(-\frac{1}{\left(1+\sin ^{2} \theta\right)^{2}}+C\)

\(\int \arctan x d x=\) (A) \(x \arctan x-\ln \left(1+x^{2}\right)+C\) (B) \(x \arctan x+\ln \left(1+x^{2}\right)+C\) (C) \(x \arctan x+\frac{1}{2} \ln \left(1+x^{2}\right)+C\) (D) \(x \arctan x-\frac{1}{2} \ln \left(1+x^{2}\right)+C\)

\(\int \frac{d x}{1-e^{x}}=\) (A) \(-\ln \left|1-e^{x}\right|+C\) (B) \(x-\ln \left|1-e^{x}\right|+C\) (C) \(\frac{1}{\left(1-e^{x}\right)^{2}}+C\) (D) \(e^{-x} \ln \left|\mathrm{I}+e^{x}\right|+C\)

\(\int \frac{(2-y)^{2}}{4 \sqrt{y}} d y=\) (A) \(\frac{1}{6}(2-y)^{3} \sqrt{y}+C\) (B) \(2 \sqrt{y}-\frac{2}{3} y^{3 / 2}+\frac{8}{5} y^{5 / 2}+C\) (C) \(\ln |y|-y+2 y^{2}+C\) (D) \(2 y^{1 / 2}-\frac{2}{3} y^{3 / 2}+\frac{1}{10} y^{5 / 2}+C\)

\(\int e^{2 \ln u} d u=\) (A) \(\frac{1}{3} e^{u^{3}}+C\) (B) \(e^{u^{3} / 3}+C\) (C) \(\frac{1}{3} u^{3}+C\) (D) \(\frac{2}{u} e^{2 \ln u}+C\)

\(\int \frac{d y}{y\left(1+\ln y^{2}\right)}=\) (A) \(\frac{1}{2} \ln \left|1+\ln y^{2}\right|+C\) (B) \(-\frac{1}{\left(1+\ln y^{2}\right)^{2}}+C\) (C) \(\ln |y|+\frac{1}{2} \ln |\ln y|+C\) (D) \(\tan ^{-1}(\ln |y|)+C\)

\(\int(\tan \theta-1)^{2} d \theta=\) (A) \(\sec \theta+\theta+2 \ln |\cos \theta|+C\) (B) \(\tan \theta+2 \ln |\cos \theta|+C\) (C) \(\tan \theta-2 \sec ^{2} \theta+C\) (D) \(\tan \theta-2 \ln |\cos \theta|+C\)

\(\int \frac{d \theta}{1+\sin \theta}=\) (A) \(\sec \theta-\tan \theta+C\) (B) \(\ln (1+\sin \theta)+C\) (C) \(\ln |\sec \theta+\tan \theta|+C\) (D) \(\tan \theta-\sec \theta+C\)

A particle starting at rest at \(t=0\) moves along a line so that its acceleration at time \(t\) is \(12 t \mathrm{ft} / \mathrm{sec}^{2}\). How much distance does the particle cover during the first 3 seconds? (A) 32 feet (B) 48 feet (C) 54 feet (D) 108 feet

The equation of the curve whose slope at point \((x y)\) is \(x^{2}-2\) and which contains the point (1,-3) is (A) \(y=\frac{1}{3} x^{3}-2 x\) (B) \(y=2 x-1\) (C) \(y=\frac{1}{3} x^{3}-\frac{10}{3}\) (D) \(y=\frac{1}{3} x^{3}-2 x-\frac{4}{3}\)

A particle moves along a line with acceleration \(2+6 t\) at time \(t\). When \(t\) \(=0\), its velocity equals 3 and it is at position \(s=2\). When \(t=1\), it is at position \(s=\) (A) 5 (B) 6 (C) 7 (D) 8

Find the acceleration (in \(\mathrm{ft} / \mathrm{sec}^{2}\) ) needed to bring a particle moving with a velocity of \(75 \mathrm{ft} / \mathrm{sec}\) to a stop in 5 seconds. (A) -3 (B) -6 (C) -15 (D) -25

\(\int \frac{x^{2}}{x^{2}-1} d x=\) (A) \(x+\frac{1}{2} \ln \left|\frac{x-1}{x+1}\right|+C\) (B) \(\ln \left|x^{2}-1\right|+C\) (C) \(x+\frac{1}{2} \ln \left|\frac{x+1}{x-1}\right|+C\) (D) \(1+\frac{1}{2} \ln \left|\frac{x+1}{x-1}\right|+C\)