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Câu hỏi: Nếu $\int\limits_{-1}^{4}{f\left( x \right)}dx=3$ và $\int\limits_{-1}^{0}{f\left( x \right)}dx=2$ thì $\int\limits_{0}^{4}{\left[ 4{{e}^{2x}}+3f\left( x \right) \right]}dx$ bằng
A. $2{{e}^{8}}+2$.
B. $2{{e}^{8}}$.
C. $2{{e}^{8}}+1$.
D. $4{{e}^{8}}-1$.
A. $2{{e}^{8}}+2$.
B. $2{{e}^{8}}$.
C. $2{{e}^{8}}+1$.
D. $4{{e}^{8}}-1$.
Ta có: $\int\limits_{-1}^{4}{f\left( x \right)}dx=3\Leftrightarrow \int\limits_{-1}^{0}{f\left( x \right)}dx+\int\limits_{0}^{4}{f\left( x \right)dx=3}$ $\Leftrightarrow \int\limits_{0}^{4}{f\left( x \right)}dx=3-\int\limits_{-1}^{0}{f\left( x \right)dx}=3-2=1$
Suy ra: $\int\limits_{0}^{4}{\left[ 4{{e}^{2x}}+3f\left( x \right) \right]}dx=4\int\limits_{0}^{4}{{{e}^{2x}}dx}+3\int\limits_{0}^{4}{f\left( x \right)}dx=2\left. {{e}^{2x}} \right|_{0}^{4}+3.1$ $=2{{e}^{8}}+1$
Suy ra: $\int\limits_{0}^{4}{\left[ 4{{e}^{2x}}+3f\left( x \right) \right]}dx=4\int\limits_{0}^{4}{{{e}^{2x}}dx}+3\int\limits_{0}^{4}{f\left( x \right)}dx=2\left. {{e}^{2x}} \right|_{0}^{4}+3.1$ $=2{{e}^{8}}+1$
Đáp án C.
