Tìm nguyên hàm \(\int \sin \sqrt{x} d x\).

\(\text { Đặt } t=\sqrt{x}, \text { ta có } \int \sin \sqrt{x} \mathrm{d} x=\int 2 t \sin t \mathrm{d} t\)

Đặt \(\left\{\begin{array}{l}
u=2 t \\
\mathrm{d} v=\sin t \mathrm{d} t
\end{array} \mathrm{tacó}\left\{\begin{array}{l}
\mathrm{d} u=2 \mathrm{d} t \\
v=-\cos t
\end{array}\right.\right.\)

Khi đó:

\(\int 2 t \sin t d t=-2 t \cos t+\int 2 \cos t d t=-2 t \cos t+2 \sin t+C\\
=-2 \sqrt{x} \cos \sqrt{x}+2 \sin \sqrt{x}+C\)