Biết \(\int {f\left( t \right)dt} = {t^2} + 3t + C.\) Tính \(\int {f\left( {\sin 2x} \right)\cos 2xdx} \)

* Hướng dẫn giải

Đặt \(\sin 2x = t \Rightarrow 2\cos 2xdx = dt \Leftrightarrow dx = \dfrac{1}{{2\cos 2x}}dt\)

Ta có

\(\begin{array}{l}\int {f\left( {\sin 2x} \right)\cos 2xdx}  = \int {f\left( t \right).\cos 2x.\dfrac{1}{{2\cos 2x}}dt}  = \dfrac{1}{2}\int {f\left( t \right)dx} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \dfrac{1}{2}{t^2} + \dfrac{3}{2}t + C = \dfrac{1}{2}{\sin ^2}2x + \dfrac{3}{2}\sin 2x + C\end{array}\)                                     

Chọn C