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Câu hỏi: Biết rằng $\int\limits_{0}^{1}{\dfrac{2{{x}^{2}}+3x+4}{x+1}dx}=a+b\ln 2$ với $a,b\in \mathbb{Z}$. Tính $S={{a}^{4}}+{{b}^{4}}$.
A. $S=162$.
B. $S=82$.
C. $S=337$.
D. $S=97$.
A. $S=162$.
B. $S=82$.
C. $S=337$.
D. $S=97$.
Ta có $\int\limits_{0}^{1}{\dfrac{2{{x}^{2}}+3x+4}{x+1}dx}=\int\limits_{0}^{1}{\dfrac{2x\left( x+1 \right)+\left( x+1 \right)+3}{x+1}}=\int\limits_{0}^{1}{\left( 2x+1+\dfrac{3}{x+1} \right)dx}$
$=\left. \left( {{x}^{2}}+x+3\ln \left| x+1 \right| \right) \right|_{0}^{1}=2+3\ln 2\Rightarrow \left\{ \begin{aligned}
& a=2 \\
& b=3 \\
\end{aligned} \right.\Rightarrow S=97$.
$=\left. \left( {{x}^{2}}+x+3\ln \left| x+1 \right| \right) \right|_{0}^{1}=2+3\ln 2\Rightarrow \left\{ \begin{aligned}
& a=2 \\
& b=3 \\
\end{aligned} \right.\Rightarrow S=97$.
Đáp án D.