TEST 18
1.
\(\displaystyle\int\samefrac{dx}{(x+2)(x+3)}=?\)
A) \(\ln\left|\samefrac{x+3}{x+2}\right|+c\)B) \(\ln\left|\samefrac{x+2}{x+3}\right|+c\)C) \(2\cdot\ln\left|\samefrac{x+3}{x+2}\right|+c\)D) \(2\cdot\ln\left|\samefrac{x+2}{x+3}\right|+c\)E) \(\samefrac{1}{2}\ln\left|\samefrac{x+2}{x+3}\right|+c\)
2.
\(\displaystyle\int\samefrac{dx}{9x^2-4}=?\)
A) \(\samefrac{1}{4}\ln\left|\samefrac{3x-2}{3x+2}\right|+c\)B) \(4\cdot\ln\left|\samefrac{3x-2}{3x+2}\right|+c\)C) \(\samefrac{4}{3}\ln\left|\samefrac{3x-2}{3x+2}\right|+c\)D) \(\samefrac{1}{12}\ln\left|\samefrac{3x-2}{3x+2}\right|+c\)E) \(\samefrac{3}{4}\ln\left|\samefrac{3x-2}{3x+2}\right|+c\)
3.
\(\displaystyle\int \samefrac{5x+7}{x^2+3x+2}\,dx=?\)
A) \(\ln|x^2+3x+2|+c\)B) \(2\cdot\ln|x^2+3x+2|+c\)C) \(\ln\left|\samefrac{x+1}{x+2}\right|+c\)D) \(\ln\left|\samefrac{x+2}{x+1}\right|+c\)E) \(2\cdot\ln|x+1|+3\cdot\ln|x+2|+c\)
4.
\(\displaystyle\int \samefrac{x^2+x+2}{(x^2+1)(x+1)}\,dx=?\)
A) \(2\arctan x+\ln|x+1|+c\)B) \(\arctan x+\ln|x+1|+c\)C) \(\arctan x+2\ln|x|+c\)D) \(\arctan x+\ln|x|+c\)E) \(\arctan x+\samefrac1{x+1}+c\)
5.
\(\displaystyle\int \samefrac{e^{2x}+1}{e^x}\,dx=?\)
A) \(e^x+e^{-x}+c\)B) \(e^x-e^{-x}+c\)C) \(e^x+x+c\)D) \(xe^x+c\)E) \(x+e^{-x}+c\)
6.
\(\displaystyle\int \samefrac{x+\sqrt x}{\sqrt[3] x}\,dx=?\)
A) \(\sqrt[3]{x^2}+\sqrt[6] x+c\)B) \(\sqrt[3]{x^5}+\sqrt[6]{x^7}+c\)C) \(\samefrac{5}{3}\sqrt[3]{x^5}+\samefrac{7}{6}\sqrt[6]{x^7}+c\)D) \(\samefrac{3}{5}\sqrt[3]{x^5}+\samefrac{6}{7}\sqrt[6]{x^7}+c\)E) \(\samefrac{3}{5}\sqrt[3]{x^2}+\samefrac{6}{7}\sqrt[6] x+c\)
7.
\(\displaystyle\int \samefrac{\cos(2x)}{\cos x-\sin x}\,dx=?\)
A) \(\sin x+\cos x+c\)B) \(\sin x-\cos x+c\)C) \(-\sin x+\cos x+c\)D) \(-\sin x-\cos x+c\)E) \(\ln|\cos x-\sin x|+c\)
8.
\(\displaystyle\int \samefrac{e^{3x}-e^x}{e^x-1}\,dx=?\)
A) \(e^{2x}+e^x+c\)B) \(e^{2x}-e^x+c\)C) \(\samefrac{e^{2x}}2-e^x+c\)D) \(\samefrac{e^{3x}}3+e^x+c\)E) \(\samefrac{e^{2x}}2+e^x+c\)