Cho tích phân I = tích phân từ 1 đến 2 x + ln x / ( x + 1 )^3 d x = a + b . ln 2 − c . ln 3 với a , b , c thuộc R , tỉ số c/a bằng

* Hướng dẫn giải

Đặt\(\left\{ {\begin{array}{*{20}{c}}{u = x + lnx}\\{dv = \frac{{dx}}{{{{(x + 1)}^3}}}}\end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{du = \frac{{x + 1}}{x}dx}\\{v = - \frac{1}{{2{{(x + 1)}^2}}}}\end{array}} \right.\)

Khi đó\[I = - \frac{{x + lnx}}{{2{{(x + 1)}^2}}}\left| {_1^2} \right. + \int\limits_1^2 {\frac{{x + 1}}{x}.\frac{1}{{2{{(x + 1)}^2}}}} dx\]

\[ = - \frac{{2 + \ln 2}}{{18}} + \frac{1}{8} + \frac{1}{2}\mathop \smallint \limits_1^2 \frac{{{\rm{d}}x}}{{x\left( {x + 1} \right)}}\]

\[ = - \frac{{2 + \ln 2}}{{18}} + \frac{1}{8} + \frac{1}{2}\mathop \smallint \limits_1^2 \left( {\frac{1}{x} - \frac{1}{{x + 1}}} \right){\rm{d}}x.\]

\( = - \frac{{2 + ln2}}{{18}} + \frac{1}{8} + \frac{1}{2}(ln|x| - ln|x + 1|)\left| {_1^2} \right.\)

\(\begin{array}{l} = \frac{1}{{72}} - \frac{1}{{18}}ln2 + \frac{1}{2}(ln2 - ln3 + ln2)\\ = \frac{1}{{72}} + \frac{{17}}{{18}}ln2 - \frac{1}{2}\ln 3\\ = a + b.ln2 - c.ln3\end{array}\)

Vậy\(\left\{ {\begin{array}{*{20}{c}}{a = \frac{1}{{72}}}\\{b = \frac{{17}}{{18}}}\\{c = \frac{1}{2}}\end{array}} \right. \Rightarrow \frac{c}{a} = \frac{1}{2}:\frac{1}{{72}} = 36\)

Đáp án cần chọn là: D