Cho hàm số $y=f\left( x \right)$ thỏa mãn $f\left( {{x}^{3}}+5x+1...

Đặt $\left\{ \begin{aligned}
& u=x \\
& dv=f'\left( x \right)dx \\
\end{aligned} \right.\Rightarrow \left\{ \begin{aligned}
& du=dx \\
& v=f\left( x \right) \\
\end{aligned} \right.$$\Rightarrow I=\left. xf\left( x \right) \right|_{1}^{7}-\int_{1}^{7}{f\left( x \right)dx}f\left( {{x}^{3}}+5x+1 \right)=5x+2\Rightarrow \left\{ \begin{aligned}
& f\left( 7 \right)=7 \left( x=1 \right) \\
& f\left( 1 \right)=2 \left( x=0 \right) \\
\end{aligned} \right.I=47-\int_{1}^{7}{f\left( x \right)dx}t={{x}^{3}}+5x+1\Rightarrow \left\{ \begin{aligned}
& dt=(3{{x}^{2}}+5)dx \\
& f\left( t \right)=5x+2 \\
\end{aligned} \right.\begin{matrix}
t=1\Rightarrow x=0 \\
t=7\Rightarrow x=1 \\
\end{matrix}I=47-\int_{1}^{7}{f\left( x \right)dx}=47-\int_{0}^{1}{\left( 5x+2 \right)\left( 3{{x}^{2}}+5 \right)dx}=18,75$.