\(\text { Cho hàm số } F(x)=\int x \sqrt{x^{2}+2} \mathrm{~d} x \text { . Biết } F(\sqrt{2})=\frac{2}{3}, \text { tính } F(\sqrt{7}) \text { . }\)

\(\begin{array}{l}
\text { Ta có: } F(x)=\int x \sqrt{x^{2}+2} \mathrm{~d} x=\frac{1}{2} \int \sqrt{x^{2}+2} \mathrm{~d}\left(x^{2}+2\right)=\frac{1}{3}\left(\sqrt{x^{2}+2}\right)^{3}+C \\
\text { Mà } F(\sqrt{2})=\frac{2}{3} \Leftrightarrow \frac{8}{3}+C=\frac{2}{3} \Leftrightarrow C=-2 \\
\text { Vậy } F(\sqrt{7})=9-2=7 .
\end{array}\)