TEST 19
1.
\(u=f(x)\)
\(v=g(x)\)
\((uv)'=u'v+v'u\)
Which of the following is correct?
A) \(\displaystyle\int u\,dv=uv+\displaystyle\int v\,du\)B) \(\displaystyle\int u\,dv=uv-\displaystyle\int v\,du\)C) \(uv=\displaystyle\int u\,du+\displaystyle\int v\,dv\)D) \(\samefrac vu=\displaystyle\int u\,dv+\displaystyle\int v\,du\)E) \(\samefrac uv=\displaystyle\int u\,dv+\displaystyle\int v\,du\)
2.
\(\displaystyle\int (xe^x)\,dx=?\)
A) \(xe^x+e^x+c\)B) \(xe^{-x}+e^{-x}+c\)C) \(xe^x-e^x+c\)D) \(xe^{-x}-e^{-x}+c\)E) \(x^2e^x+e^x+c\)
3.
\(\displaystyle\int (x\cdot\sin x)\,dx=?\)
A) \(x\cdot\cos x-\sin x+c\)B) \(-x\cdot\cos x-\sin x+c\)C) \(x\cdot\cos x+\sin x+c\)D) \(-x\cdot\cos x+\cos x+c\)E) \(-x\cdot\cos x+\sin x+c\)
4.
\(\displaystyle\int (\ln x)\,dx=?\)
A) \(x\cdot\ln|x|+x+c\)B) \(x\cdot\ln|x|-2x+c\)C) \(x\cdot\ln|x|+2x+c\)D) \(\ln|x|+x+c\)E) \(x\cdot\ln|x|-x+c\)
5.
\(\displaystyle\int (x\cdot\ln x)\,dx=?\)
A) \(x\ln x-\samefrac{x^2}{4}+c\)B) \(x\ln x-\samefrac{x^2}{2}+c\)C) \(\samefrac{x^2}{2}\ln x-\samefrac{x^2}{2}+c\)D) \(\samefrac{x^2}{2}\ln x+\samefrac{x^2}{4}+c\)E) \(\samefrac{x^2}{2}\ln x-\samefrac{x^2}{4}+c\)
6.
\(\displaystyle\int (x^2\cdot e^x)\,dx=?\)
A) \(e^x\cdot(x^2-2x+2)+c\)B) \(e^x\cdot(x^2+2x+2)+c\)C) \(e^x\cdot(x^2-2x+1)+c\)D) \(e^x\cdot(x^2-2x-2)+c\)E) \(e^x\cdot(x^2+x+1)+c\)
7.
\(\displaystyle\int (x^3\sin x)\,dx=?\)
A) \(-x^3\cdot\cos x+3x^2\cdot\sin x+c\)B) \(6x\cdot\cos x-6\cdot\sin x+c\)C) \(-x^3\cdot\cos x+6\cdot x\cos x+c\)D) \(3x^2\cdot\sin x-6\cdot\sin x+c\)E) \(-x^3\cdot\cos x+3x^2\cdot\sin x+6x\cdot\cos x-6\cdot\sin x+c\)
8.
\(\displaystyle\int \ln(3x+2)\,dx=?\)
A) \(x\cdot\ln|3x+2|+c\)B) \(\samefrac{x\cdot\ln|3x+2|}3+c\)C) \(\samefrac{x\cdot\ln|3x+2|}3+3x+c\)D) \((3x+2)\cdot\ln|3x+2|+c\)E) \(\samefrac{3x+2}3\cdot\ln|3x+2|-x+c\)