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Câu hỏi: Biết $\int\limits_{0}^{1}{{{2}^{x}}f\left( {{2}^{x}} \right)}\text{d}x=\text{lo}{{\text{g}}_{2}}3$. Khi đó $\int\limits_{1}^{2}{f\left( x \right)}\text{d}x$ bằng
A. $\ln 3$.
B. ${{\log }_{3}}\text{e}$.
C. ${{\log }_{2}}9$.
D. ${{\log }_{2}}\sqrt{3}$.
A. $\ln 3$.
B. ${{\log }_{3}}\text{e}$.
C. ${{\log }_{2}}9$.
D. ${{\log }_{2}}\sqrt{3}$.
Đặt $t={{2}^{x}}\Rightarrow \text{d}t={{2}^{x}}\text{ln}2\text{d}x$.
Đổi cận: $x=0\to t=1$, $x=1\to t=2$.
Khi đó: $\int\limits_{0}^{1}{{{2}^{x}}f\left( {{2}^{x}} \right)}\text{d}x=\dfrac{1}{\text{ln}2}\int\limits_{1}^{2}{f\left( t \right)}\text{d}t=\text{lo}{{\text{g}}_{2}}3\Leftrightarrow \int\limits_{1}^{2}{f\left( t \right)}\text{d}t=\text{ln}2.\text{lo}{{\text{g}}_{2}}3=\text{ln}3$.
Vậy $\int\limits_{1}^{2}{f\left( x \right)}\text{d}x=\text{ln}3$.
Đổi cận: $x=0\to t=1$, $x=1\to t=2$.
Khi đó: $\int\limits_{0}^{1}{{{2}^{x}}f\left( {{2}^{x}} \right)}\text{d}x=\dfrac{1}{\text{ln}2}\int\limits_{1}^{2}{f\left( t \right)}\text{d}t=\text{lo}{{\text{g}}_{2}}3\Leftrightarrow \int\limits_{1}^{2}{f\left( t \right)}\text{d}t=\text{ln}2.\text{lo}{{\text{g}}_{2}}3=\text{ln}3$.
Vậy $\int\limits_{1}^{2}{f\left( x \right)}\text{d}x=\text{ln}3$.
Đáp án A.
