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Câu hỏi: Cho $\int{f}(4x)dx={{x}^{2}}+3x+c$. Mệnh đề nào sau đây đúng ?
A. $\int{f}(x+2)dx=\dfrac{{{x}^{2}}}{4}+2x+C$.
B. $\int{f}(x+2)dx={{x}^{2}}+7x+C$.
C. $\int{f}(x+2)dx=\dfrac{{{x}^{2}}}{4}+4x+C$.
D. $\int{f}(x+2)dx=\dfrac{{{x}^{2}}}{2}+4x+C$.
Từ $\int{f}(4x)dx={{x}^{2}}+3x+c\Rightarrow {{\left( \int{f}(4x)dx \right)}^{\prime }}={{\left( {{x}^{2}}+3x+c \right)}^{\prime }}\Rightarrow f\left( 4x \right)=2x+3$
Đặt $4x=t+2\Rightarrow x=\dfrac{t+2}{4}$
Do đó $f\left( t+2 \right)=\dfrac{t}{2}+4\Rightarrow f\left( x+2 \right)=\dfrac{x}{2}+4\Rightarrow \int{f}(x+2)dx=\dfrac{{{x}^{2}}}{4}+4x+C$
A. $\int{f}(x+2)dx=\dfrac{{{x}^{2}}}{4}+2x+C$.
B. $\int{f}(x+2)dx={{x}^{2}}+7x+C$.
C. $\int{f}(x+2)dx=\dfrac{{{x}^{2}}}{4}+4x+C$.
D. $\int{f}(x+2)dx=\dfrac{{{x}^{2}}}{2}+4x+C$.
Từ $\int{f}(4x)dx={{x}^{2}}+3x+c\Rightarrow {{\left( \int{f}(4x)dx \right)}^{\prime }}={{\left( {{x}^{2}}+3x+c \right)}^{\prime }}\Rightarrow f\left( 4x \right)=2x+3$
Đặt $4x=t+2\Rightarrow x=\dfrac{t+2}{4}$
Do đó $f\left( t+2 \right)=\dfrac{t}{2}+4\Rightarrow f\left( x+2 \right)=\dfrac{x}{2}+4\Rightarrow \int{f}(x+2)dx=\dfrac{{{x}^{2}}}{4}+4x+C$
Đáp án C.