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Câu hỏi: Xét $\int\limits_{0}^{\dfrac{\pi }{2}}{{{\cos }^{7}}x\sin x}\text{d}x$ bằng cách đặt $t=\cos x$, mệnh đề nào dưới đây đúng?
A. $\int\limits_{0}^{\dfrac{\pi }{2}}{{{t}^{7}}}\text{d}t$.
B. $-\int\limits_{0}^{\dfrac{\pi }{2}}{{{t}^{7}}}\text{d}t$.
C. $\int\limits_{0}^{1}{{{t}^{7}}}\text{d}t$.
D. $-\int\limits_{0}^{1}{{{t}^{7}}}\text{d}t$.
A. $\int\limits_{0}^{\dfrac{\pi }{2}}{{{t}^{7}}}\text{d}t$.
B. $-\int\limits_{0}^{\dfrac{\pi }{2}}{{{t}^{7}}}\text{d}t$.
C. $\int\limits_{0}^{1}{{{t}^{7}}}\text{d}t$.
D. $-\int\limits_{0}^{1}{{{t}^{7}}}\text{d}t$.
Đặt $t=\cos x\Rightarrow \text{d}t=-\sin x\text{d}x$.
Với $x=0\Rightarrow t=1;x=\dfrac{\pi }{2}\Rightarrow t=0$.
Khi đó $\int\limits_{0}^{\dfrac{\pi }{2}}{{{\cos }^{7}}x\sin x}\text{d}x=-\int\limits_{1}^{0}{{{t}^{7}}}\text{d}t=\int\limits_{0}^{1}{{{t}^{7}}}\text{d}t$.
Với $x=0\Rightarrow t=1;x=\dfrac{\pi }{2}\Rightarrow t=0$.
Khi đó $\int\limits_{0}^{\dfrac{\pi }{2}}{{{\cos }^{7}}x\sin x}\text{d}x=-\int\limits_{1}^{0}{{{t}^{7}}}\text{d}t=\int\limits_{0}^{1}{{{t}^{7}}}\text{d}t$.
Đáp án C.
