Cho hàm số \(f(x)\) liên tục trong đoạn \(\left[ 1;e \right]\), biết \(\int\limits_{1}^{e}{\frac{f(x)}{x}dx}=1,\,\,f(e)=2\). Tích phân \(\int\limits_{1}^{e}{f'(x)\ln xdx}=?\)

\(\frac{{19}}{3}\).

\(\frac{{32}}{3}\).

\(\frac{{16}}{3}\).

\(\frac{{21}}{2}\).

$\int\limits_a^b {kf\left( x \right)dx} {\rm{\;}} = k\int\limits_a^b {f\left( x \right)dx} $

$\int\limits_a^b {xf\left( x \right)dx} {\rm{\;}} = x\int\limits_a^b {f\left( x \right)dx} $

$\int\limits_a^b {f\left( x \right)dx} {\rm{\;}} = {\rm{\;}} - \int\limits_b^a {f\left( x \right)dx} $

$\int\limits_a^b {\left[ {f\left( x \right) + g\left( x \right)} \right]dx} {\rm{\;}} = \int\limits_a^b {f\left( x \right)dx} {\rm{\;}} + \int\limits_a^b {g\left( x \right)dx} $

\(I =  - 12\)                          

\(I = 8\)                    

\(I = 12\)

\(I =  - 8\)  

$I = 9.$

$I = 4.$

$I =  - \,11.$

$I = 11.$

\(f\left( x \right) =  - \dfrac{{{\pi ^x}}}{{\ln \pi }}\).

\(f(x) = \dfrac{{{\pi ^x}}}{{\ln \pi }}\).

\(f\left( x \right) = {\pi ^x}.\ln \pi \).

\(f\left( x \right) =  - {\pi ^x}.\ln \pi \).

$\int {f\left( {3x} \right){\mkern 1mu} {\rm{d}}x = 2x\ln \left( {9x - 1} \right) + C.} $

$\int {f\left( {3x} \right){\mkern 1mu} {\rm{d}}x = 6x\ln \left( {3x - 1} \right) + C.} $

$\int {f\left( {3x} \right){\mkern 1mu} {\rm{d}}x = 6x\ln \left( {9x - 1} \right) + C.} $

$\int {f\left( {3x} \right){\mkern 1mu} {\rm{d}}x = 3x\ln \left( {9x - 1} \right) + C.} $