nguyên hàm x sin x cos x d x bằng:

8/20

\[\smallint x\sin x\cos xdx\]bằng:

\[\frac{1}{2}\left( {\frac{1}{4}\sin 2x - \frac{x}{2}\cos 2x} \right) + C\]

\[ - \frac{1}{2}\left( {\frac{1}{2}\sin 2x - \frac{x}{4}\cos 2x} \right) + C\]

\[\frac{1}{2}\left( {\frac{1}{2}\sin 2x + \frac{x}{2}\cos 2x} \right) + C\]

\[ - \frac{1}{2}\left( {\frac{1}{2}\sin 2x + \frac{x}{4}\cos 2x} \right) + C\]

Giải thích

\[I = \smallint x\sin x\cos xdx = \frac{1}{2}\smallint x\sin 2xdx\]

Đặt\(\left\{ {\begin{array}{*{20}{c}}{u = x}\\{dv = sin2xdx}\end{array}} \right. \Rightarrow \left\{ {\begin{array}{*{20}{c}}{du = dx}\\{v = - \frac{{cos2x}}{2}}\end{array}} \right.\)

\[ \Rightarrow I = \frac{1}{2}\left( { - x.\frac{{\cos 2x}}{2} + \frac{1}{2}\smallint \cos 2xdx} \right) + C\]

\[ = \frac{1}{2}\left( { - \frac{{x\cos 2x}}{2} + \frac{{\sin 2x}}{4}} \right) + C\]

Đáp án cần chọn là: A