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Câu hỏi: Cho $\int\limits_{1}^{8}{f\left( x \right)} \text{d}x=5$, hãy tính $I=\int\limits_{1}^{2}{{{x}^{2}}f\left( {{x}^{3}} \right)} \text{d}x.$
A. $\dfrac{5}{3}$.
B. $8$.
C. $5$.
D. $15$.
A. $\dfrac{5}{3}$.
B. $8$.
C. $5$.
D. $15$.
Đặt $t={{x}^{3}}\Rightarrow \text{d}t=3{{x}^{2}}\text{d}x$.
Đổi cận $x=1\Rightarrow t=1$ ; $x=2\Rightarrow t=8$.
Ta có $I=\int\limits_{1}^{8}{f\left( t \right)} \dfrac{\text{d}t}{3}=\dfrac{1}{3}\int\limits_{1}^{8}{f\left( t \right)} \text{d}t=\dfrac{1}{3}\int\limits_{1}^{8}{f\left( x \right)} \text{d}x=\dfrac{1}{3}.5=\dfrac{5}{3}.$
Đổi cận $x=1\Rightarrow t=1$ ; $x=2\Rightarrow t=8$.
Ta có $I=\int\limits_{1}^{8}{f\left( t \right)} \dfrac{\text{d}t}{3}=\dfrac{1}{3}\int\limits_{1}^{8}{f\left( t \right)} \text{d}t=\dfrac{1}{3}\int\limits_{1}^{8}{f\left( x \right)} \text{d}x=\dfrac{1}{3}.5=\dfrac{5}{3}.$
Đáp án A.