Cho hàm số $f\left( x \right)=\left\{ \begin{aligned} &...

Cách giải:
Xét $I=\int_{0}^{\dfrac{\pi}{2}} f(2 \sin x+1) \cos x d x$
Đặt $t=2\operatorname{s}\text{inx+1}$ ta có $dt=2\cos xdx.$
Đổi cận: $\left\{ \begin{aligned}
& x=0\Rightarrow t=1 \\
& x=\dfrac{\pi }{2}\Rightarrow t=3 \\
\end{aligned} \right..$ Khi đó ta có:
$I=\dfrac{1}{2}\int\limits_{1}^{3}{f\left( t \right)dt=\dfrac{1}{2}}\int\limits_{1}^{3}{f\left( x \right)dx}$
$=\dfrac{1}{2}\left( \int\limits_{1}^{2}{f\left( x \right)dx+\int\limits_{2}^{3}{f\left( x \right)dx}} \right)$
$=\dfrac{1}{2}\left( \int\limits_{1}^{2}{\left( {{x}^{2}}-2x+3 \right)dx+\int\limits_{2}^{3}{\left( {{x}^{2}}-1 \right)dx}} \right)$
$=\dfrac{1}{2}\left( \dfrac{7}{3}+\dfrac{16}{3} \right)=\dfrac{23}{6}$