Tìm ∫x(sinx2+cosx2)2dx

Question

Tìm \(\int x\left(\sin \frac{x}{2} +\cos \frac{x}{2} \right)^{2} {\rm d}x \)

 A. \(\frac{x^{2} }{2} -x\cos x+\sin x+C.\) B. \(\frac{x^{2} }{2} +x\cos x+\sin x+C. \)

 C. \(\frac{x^{2} }{2} +x\cos x-\sin x+C. \) D. \(\frac{x^{2} }{2} -x\cos x-\sin x+C.\)

Answer 1

Chọn A

 Ta có: \(\int x\left(\sin \frac{x}{2} +\cos \frac{x}{2} \right)^{2} {\rm d}x =\int x\left(\sin ^{2} \frac{x}{2} +2\sin ^{2} \frac{x}{2} \cos \frac{x}{2} +\cos ^{2} \frac{x}{2} \right){\rm d}x =\int x\left(1+\sin x\right){\rm d}x \)
\(=\int \left(x+x\sin x\right){\rm d}x =\frac{x^{2} }{2} -x\cos x+\int \cos x{\rm d}x =\frac{x^{2} }{2} -x\cos x+\sin x+C.\)