TEST 36
1.
\(\displaystyle\int\samefrac{3\,dx}{e^x+3}=?\)
A) \(\ln|e^x-3|+c\)B) \(\ln|e^x+3|+c\)C) \(x-\ln|e^x+3|+c\)D) \(x+\ln|e^x+3|+c\)E) \(x-e^x+c\)
2.
\(\displaystyle\int\samefrac{dx}{x^2+4x+5}=?\)
A) \(\arctan x+c\)B) \(\arctan(x-1)+c\)C) \(\arctan(x+2)+c\)D) \(\arctan(x-2)+c\)E) \(2\cdot\arctan(x+2)+c\)
3.
\(\displaystyle\int \samefrac{9x}{1+9x^4}\,dx=?\)
A) \(\arctan(\ln|3x|)+c\)B) \(\arctan(2x^2)+c\)C) \(\samefrac{3}{2}\arctan(3x^2)+c\)D) \(\arctan(3x)+c\)E) \(\arctan(\ln(2x))+c\)
4.
\(\displaystyle\int \samefrac{x+2}{x-1}\,dx=?\)
A) \(x+3\cdot\ln|x-1|+c\)B) \(x+\ln|x-1|+c\)C) \(\ln\left|\samefrac{x+2}{x-1}\right|+c\)D) \(\ln\left|\samefrac x{x-1}\right|+c\)E) \(2\cdot\ln|x-2|+x+c\)
5.
\(\displaystyle\int \samefrac{x-1}{x^2-2x+6}\,dx=?\)
A) \(\ln(x^2-2x+6)+c\)B) \(x^2-2x+c\)C) \(2\cdot\ln(x^2-2x+6)+c\)D) \(\ln\left|\samefrac{x-1}{x^2-2x+6}\right|+c\)E) \(\samefrac{\ln(x^2-2x+6)}2+c\)
6.
What is the answer if \(\displaystyle\int \samefrac{\sqrt[3]{x-1}+2}{\sqrt{x-1}}\,dx\) integral is converted to \(x=t^6+1\)?
A) \(6\displaystyle\int (t^4+t)\,dt\)B) \(6\displaystyle\int (t^4+t^2)\,dt\)C) \(3\displaystyle\int \samefrac{t^2+2}{t^3}\,dt\)D) \(6\displaystyle\int (t^4+2t^2)\,dt\)E) \(3\displaystyle\int (t^2+1)\,dt\)
7.
\(\begin{gathered}f(x)=x-5\\g(x)=3x^2+1\end{gathered}\)
\(\Rightarrow \displaystyle\int\limits_{-2}^{0} (f\circ g)(x)\,dx=?\)
A) \(-16\)B) \(-8\)C) \(0\)D) \(8\)E) \(16\)
8.
\(\Rightarrow\text{Shaded Area}=?\)
A) \(\samefrac{1}{2}\)B) \(1\)C) \(\samefrac{3}{2}\)D) \(2\)E) \(\samefrac{5}{2}\)