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Câu hỏi: Biết $\int\limits_{-1}^{4}{f\left( x \right)\text{d}x}=3$ và $\int\limits_{-1}^{0}{f\left( x \right)\text{d}x}=2$, tính $\int\limits_{0}^{4}{\left[ 4{{\text{e}}^{2x}}+3f\left( x \right) \right]\text{d}x}$.
A. $2{{\text{e}}^{8}}$.
B. $\text{4}{{\text{e}}^{8}}-1$.
C. $2{{\text{e}}^{8}}+1$.
D. $2{{\text{e}}^{8}}+2$.
A. $2{{\text{e}}^{8}}$.
B. $\text{4}{{\text{e}}^{8}}-1$.
C. $2{{\text{e}}^{8}}+1$.
D. $2{{\text{e}}^{8}}+2$.
Ta có $\int\limits_{-1}^{4}{f\left( x \right)\text{d}x}=\int\limits_{-1}^{0}{f\left( x \right)\text{d}x}+\int\limits_{0}^{4}{f\left( x \right)\text{d}x}$ nên $\int\limits_{0}^{4}{f\left( x \right)\text{d}x}=1$.
Do đó $\int\limits_{0}^{4}{\left[ 4{{\text{e}}^{2x}}+3f\left( x \right) \right]\text{d}x}=4\int\limits_{0}^{4}{{{\text{e}}^{2x}}\text{d}x}+3\int\limits_{0}^{4}{f\left( x \right)\text{d}x}=\left. 2{{\text{e}}^{2x}} \right|_{0}^{4}+3.1=2{{\text{e}}^{8}}+1$.
Do đó $\int\limits_{0}^{4}{\left[ 4{{\text{e}}^{2x}}+3f\left( x \right) \right]\text{d}x}=4\int\limits_{0}^{4}{{{\text{e}}^{2x}}\text{d}x}+3\int\limits_{0}^{4}{f\left( x \right)\text{d}x}=\left. 2{{\text{e}}^{2x}} \right|_{0}^{4}+3.1=2{{\text{e}}^{8}}+1$.
Đáp án C.