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Câu hỏi: Xét tích phân $I=\int\limits_{1}^{{{e}^{2}}}{\dfrac{{{\left( 1+2\ln x \right)}^{2}}}{x}\text{d}x}$ nếu đặt $t=1+2\ln x$ thì $I$ bằng
A. $\dfrac{1}{2}\int\limits_{1}^{{{e}^{2}}}{{{t}^{2}}\text{d}t}$.
B. $2\int\limits_{1}^{5}{{{t}^{2}}\text{d}t}$.
C. $2\int\limits_{1}^{{{e}^{2}}}{{{t}^{2}}\text{d}t}$.
D. $\dfrac{1}{2}\int\limits_{1}^{5}{{{t}^{2}}\text{d}t}$.
A. $\dfrac{1}{2}\int\limits_{1}^{{{e}^{2}}}{{{t}^{2}}\text{d}t}$.
B. $2\int\limits_{1}^{5}{{{t}^{2}}\text{d}t}$.
C. $2\int\limits_{1}^{{{e}^{2}}}{{{t}^{2}}\text{d}t}$.
D. $\dfrac{1}{2}\int\limits_{1}^{5}{{{t}^{2}}\text{d}t}$.
Đặt $t=1+2\ln x$ $\Rightarrow \text{d}t=\dfrac{2\text{d}x}{x}$ $\Rightarrow \dfrac{\text{d}t}{2}=\dfrac{\text{d}x}{x}$.
Đổi cận: $\left\{ \begin{aligned}
& x=1\Rightarrow t=1 \\
& x={{e}^{2}}\Rightarrow t=5 \\
\end{aligned} \right.$
Khi đó $I=\int\limits_{1}^{{{e}^{2}}}{\dfrac{{{\left( 1+2\ln x \right)}^{2}}}{x}\text{d}x}=\int\limits_{1}^{5}{\dfrac{{{t}^{2}}}{2}\text{d}t}=\dfrac{1}{2}\int\limits_{1}^{5}{{{t}^{2}}\text{d}t}$.
Đổi cận: $\left\{ \begin{aligned}
& x=1\Rightarrow t=1 \\
& x={{e}^{2}}\Rightarrow t=5 \\
\end{aligned} \right.$
Khi đó $I=\int\limits_{1}^{{{e}^{2}}}{\dfrac{{{\left( 1+2\ln x \right)}^{2}}}{x}\text{d}x}=\int\limits_{1}^{5}{\dfrac{{{t}^{2}}}{2}\text{d}t}=\dfrac{1}{2}\int\limits_{1}^{5}{{{t}^{2}}\text{d}t}$.
Đáp án D.
