Câu hỏi 3

Đặt \(\left\{ \begin{array}{l}u = \ln x\\dv = {x^3}dx\end{array} \right.\)\( \Rightarrow \left\{ \begin{array}{l}du = \dfrac{1}{x}dx\\v = \dfrac{{{x^4}}}{4}\end{array} \right.\)

Khi đó \(\int\limits_1^e {{x^3}} \ln xdx = \left. {\ln x.\dfrac{{{x^4}}}{4}} \right|_1^e - \dfrac{1}{4}\int\limits_1^e {{x^3}dx}  = \dfrac{{{e^4}}}{4} - \dfrac{1}{4}.\dfrac{1}{4}\left. {{x^4}} \right|_1^e = \dfrac{{{e^4}}}{4} - \dfrac{1}{{16}}\left( {{e^4} - 1} \right) = \dfrac{{3{e^4} + 1}}{{16}}\).

Do đó \(a = 4,b = 16 \Rightarrow ab = 64\).

Chọn A