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Câu hỏi: Nguyên hàm $F\left( x \right)$ của hàm số $f\left( x \right)=\sin x+\cos x$ thỏa mãn $F\left( \dfrac{\pi }{2} \right)=2$
A. $F\left( x \right)=\cos x-\sin x+3$
B. $F\left( x \right)=-\cos x+\sin x+3$
C. $F\left( x \right)=-\cos x+\sin x-1$
D. $F\left( x \right)=-\cos x+\sin x+1$
A. $F\left( x \right)=\cos x-\sin x+3$
B. $F\left( x \right)=-\cos x+\sin x+3$
C. $F\left( x \right)=-\cos x+\sin x-1$
D. $F\left( x \right)=-\cos x+\sin x+1$
Có $F\left( x \right)=\int{f\left( x \right)dx}=\int{\left( \sin x+\cos x \right)dx}=-\cos x+\sin x+C$
Do $F\left( \dfrac{\pi }{2} \right)=-\cos \dfrac{\pi }{2}+\sin \dfrac{\pi }{2}+C=2\Leftrightarrow 1+C=2\Leftrightarrow C=1$
$\Rightarrow F\left( x \right)=-\cos x+\sin x+1$
Do $F\left( \dfrac{\pi }{2} \right)=-\cos \dfrac{\pi }{2}+\sin \dfrac{\pi }{2}+C=2\Leftrightarrow 1+C=2\Leftrightarrow C=1$
$\Rightarrow F\left( x \right)=-\cos x+\sin x+1$
Đáp án D.