Chapter 5

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

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

\(\int \sqrt{4-2 t} \mathrm{~d} t=\) (A) \(-\frac{1}{3}(4-2 t)^{3 / 2}+C\) (B) \(\frac{2}{3}(4-2 t)^{3 / 2}+C\) (C) \(-\frac{1}{6}(4-2 t)^{3}+C\) (D) \(\frac{4}{3}(4-2 t)^{3 / 2}+C\)

\(\int(2-3 x)^{5} d x=\) (A) \(\frac{1}{6}(2-3 x)^{6}+C\) (B) \(-\frac{1}{2}(2-3 x)^{6}+C\) (C) \(\frac{1}{2}(2-3 x)^{6}+C\) (D) \(-\frac{1}{18}(2-3 x)^{6}+C\)

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

\(\int \frac{d x}{3(2 x-1)^{2}}=\) (A) \(\frac{-3}{2 x-1}+C\) (B) \(\frac{1}{6-12 x}+C\) (C) \(\frac{6}{2 x-1}+C\) (D) \(\frac{1}{3} \ln |2 x-1|+C\)

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

\(\int \frac{t}{\sqrt{2 t^{2}-1}} d t=\) (A) \(\frac{1}{2} \ln \sqrt{2 t^{2}-1}+C\) (B) \(8 \sqrt{2 t^{2}-1}+C\) (C) \(-\frac{1}{4\left(2 t^{2}-1\right)}+C\) (D) \(\frac{1}{2} \sqrt{2 t^{2}-1}+C\)

\(\int \cos 3 x d x=\) (A) \(3 \sin 3 x+C\) (B) \(-\sin 3 x+C\) (C) \(-\frac{1}{3} \sin 3 x+C\) (D) \(\frac{1}{3} \sin 3 x+C\)

\(\int \frac{x d x}{1+4 x^{2}}=\) (A) \(\frac{1}{8} \ln \left(1+4 x^{2}\right)+C\) (B) \(\frac{1}{4} \sqrt{1+4 x^{2}}+C\) (C) \(\frac{1}{2} \ln \left|1+4 x^{2}\right|+C\) (D) \(\frac{1}{2} \tan ^{-1} 2 x+C\)

\(\int \frac{d x}{1+4 x^{2}}=\) (A) \(\tan ^{-1}(2 x)+C\) (B) \(\frac{1}{8} \ln \left(1+4 x^{2}\right)+C\) (C) \(\frac{1}{2} \tan ^{-1}(2 x)+C\) (D) \(\frac{1}{8 x} \ln \left|1+4 x^{2}\right|+C\)

\(\int \frac{x}{\left(1+4 x^{2}\right)^{2}} d x=\) (A) \(\frac{1}{8} \ln \left(1+4 x^{2}\right)^{2}+C\) (B) \(-\frac{1}{8\left(1+4 x^{2}\right)}+C\) (C) \(-\frac{1}{3\left(1+4 x^{2}\right)^{3}}+C\) (D) \(-\frac{1}{\left(1+4 x^{2}\right)}+C\)

\(\int \frac{x d x}{\sqrt{1+4 x^{2}}}=\) (A) \(\frac{1}{8} \sqrt{1+4 x^{2}}+C\) (B) \(\frac{\sqrt{1+4 x^{2}}}{4}+C\) (C) \(\frac{1}{2} \sin ^{-1} 2 x+C\) (D) \(\frac{1}{8} \ln \sqrt{1+4 x^{2}}+C\)

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

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

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

\(\int \frac{(x-2)^{3}}{x^{2}} d x=\) (A) \(\frac{(x-2)^{4}}{4 x^{2}}+C\) (B) \(\frac{x^{2}}{2}-6 x+6 \ln |x|-\frac{8}{x}+C\) (C) \(\frac{x^{2}}{2}-3 x+6 \ln |x|+\frac{4}{x}+C\) (D) \(\frac{x^{2}}{2}-6 x+12 \ln |x|+\frac{8}{x}+C\)

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

\(\int\left(4 x^{1 / 3}-5 x^{3 / 2}-x^{-1 / 2}\right) d x=\) (A) \(3 x^{4 / 3}-2 x^{5 / 2}-2 x^{1 / 2}+C\) (B) \(3 x^{4 / 3}-2 x^{5 / 2}+2 x^{1 / 2}+C\) (C) \(6 x^{2 / 3}-2 x^{5 / 2}-\frac{1}{2} x^{2}+C\) (D) \(\frac{16}{9} x^{4 / 3}-\frac{25}{2} x^{5 / 2}-\frac{1}{2} x^{1 / 2}+C\)

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

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

\(\int \frac{u d u}{\sqrt{4-9 u^{2}}}=\) (A) \(\frac{1}{3} \sin ^{-1} \frac{3 u}{2}+C\) (B) \(-\frac{1}{18} \ln \sqrt{4-9 u^{2}}+C\) (C) \(2 \sqrt{4-9 u^{2}}+C\) (D) \(-\frac{1}{9} \sqrt{4-9 u^{2}}+C\)

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

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

\(\int t \cos \left(4 t^{2}\right) d t=\) (A) \(\frac{1}{8} \sin \left(4 t^{2}\right)+C\) (B) \(\frac{1}{2} \cos ^{2}(2 t)+C\) (C) \(-\frac{1}{8} \sin \left(4 t^{2}\right)+C\) (D) \(\frac{1}{4} \sin \left(4 t^{2}\right)+C\)

\(\int \cos ^{2} 2 x d x=\) (A) \(\frac{x}{2}+\frac{\sin 4 x}{8}+C\) (B) \(\frac{x}{2}-\frac{\sin 4 x}{8}+C\) (C) \(\frac{x}{4}+\frac{\sin 4 x}{4}+C\) (D) \(\frac{x}{4}+\frac{\sin 4 x}{16}+C\)

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

\(\int x \cos x d x=\) (A) \(x \sin x+C\) (B) \(x \sin x+\cos x+C\) (C) \(x \sin x-\cos x+C\) (D) \(\cos x-x \sin x+C\)

\(\int \frac{d u}{\cos ^{2} 3 u}=\) (A) \(-\frac{\sec 3 u}{3}+C\) (B) \(\tan 3 u+C\) (C) \(u+\frac{\sec 3 u}{3}+C\) (D) \(\frac{1}{3} \tan 3 u+C\)

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

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

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

\(\int \sec ^{3 / 2} x \tan x d x=\) (A) \(\frac{2}{5} \sec ^{5 / 2} x+C\) (B) \(-\frac{2}{3} \cos ^{-3 / 2} x+C\) (C) \(\sec ^{3 / 2} x+C\) (D) \(\frac{2}{3} \sec ^{3 / 2} x+C\)

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

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

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

\(\int \sin 2 \theta \cos \theta d \theta=\) (A) \(-\frac{2}{3} \cos ^{3} \theta+C\) (B) \(\frac{2}{3} \cos ^{3} \theta+C\) (C) \(\sin ^{2} \theta \cos \theta+C\) (D) \(\cos ^{3} \theta+C\)

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

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

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

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

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

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

\(\int e^{2 \theta} \sin e^{2 \theta} d \theta=\) (A) \(\cos e^{2 \theta}+C\) (B) \(2 e^{4 \theta}\left(\cos e^{2 \theta}+\sin e^{2 \theta}\right)+C\) (C) \(-\frac{1}{2} \cos e^{2 \theta}+C\) (D) \(-2 \cos e^{2 \theta}+C\)

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

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

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

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

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

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