TEST 16
1.
\(\displaystyle\int \samefrac5{(x-2)(x+3)}\,dx=?\)
A) \(5\cdot\ln\left|\samefrac{x-2}{x+3}\right|+c\)B) \(\samefrac{1}{5}\ln\left|\samefrac{x-2}{x+3}\right|+c\)C) \(5\cdot\ln\left|\samefrac{x+3}{x-2}\right|+c\)D) \(\samefrac{1}{5}\ln\left|\samefrac{x+3}{x-2}\right|+c\)E) \(\ln\left|\samefrac{x-2}{x+3}\right|+c\)
2.
\(\displaystyle\int\samefrac{dx}{(x+1)(x+2)}=?\)
A) \(\ln\left|\samefrac{x+2}{x+1}\right|+c\)B) \(\ln\left|\samefrac{x+1}{x+2}\right|+c\)C) \(\ln|(x+1)(x+2)|+c\)D) \(2\cdot\ln\left|\samefrac{x+1}{x+2}\right|+c\)E) \(\samefrac{1}{2}\ln\left|\samefrac{x+2}{x+1}\right|+c\)
3.
\(\displaystyle\int \samefrac{2x}{(x^2+1)(x^2+2x+1)}\,dx=?\)
A) \(\arcsin x+\samefrac1{1-x}+c\)B) \(\arcsin x-\samefrac1{1-x}+c\)C) \(\operatorname{arccot}x-\samefrac1{1-x}+c\)D) \(\arctan x+\samefrac1{x+1}+c\)E) \(\ln|x^2+2x+1|+c\)
4.
\(\displaystyle\int \samefrac{1+\sin(2x)}{\sin x+\cos x}\,dx=?\)
A) \(-\cos x+\sin x+c\)B) \(\cos x-\sin x+c\)C) \(\ln\left|\samefrac{\cos x}{\sin x}\right|+c\)D) \(\ln|\sin(2x)|+c\)E) \(\samefrac{(1+\sin^2x)^2}2+c\)
5.
\(\displaystyle\int \samefrac{2x-1}{(x^2+1)(x+2)}\,dx=?\)
A) \(\samefrac{1}{2}\ln|1+x^2|-\ln|x+2|+c\)B) \(\ln\left|\samefrac{1+x^2}{x+2}\right|+c\)C) \(\ln|(1+x^2)(x+2)|+c\)D) \(2\cdot\ln|(x+2)(x^2+1)|+c\)E) \(\arctan x+\ln|x+2|+c\)
6.
\(\displaystyle\int \samefrac{e^{2x}-1}{e^x+1}\,dx=?\)
A) \(e^x+e^{-x}+c\)B) \(e^{2x}+e^x+c\)C) \(e^x+x+c\)D) \(e^x-x+c\)E) \(e^x+c\)
7.
\(\displaystyle\int \samefrac{2x+1}{x^2+1}\,dx=?\)
A) \(\arctan x+c\)B) \(\arctan x+\ln|x+1|+c\)C) \(\arctan x+\ln|1+x^2|+c\)D) \(\arcsin x+\ln|1+x^2|+c\)E) \(\arcsin x+\ln|1-x^2|+c\)
8.
\(\displaystyle\int \samefrac{x-1}{x^2+1}\,dx=?\)
A) \(\ln|x+1|+c\)B) \(x+\ln|x+1|+c\)C) \(\ln\left|\samefrac{x+1}{x-1}\right|+c\)D) \(\samefrac{1}{2}\ln|1+x^2|+\arctan x+c\)E) \(\samefrac{1}{2}\ln|1+x^2|+\operatorname{arccot}x+c\)