Cho \(F\left( x \right)\) là một nguyên hàm của hàm số \(f\left( x \right)=\left| 1+x \right|-\left| 1-x \right|\) trên tập \(\mathbb{R}\) và thỏa mãn \(F\left( 1 \right)=3;\ \ F\left( -1 \right)=2;\ \ F\left( -2 \right)=4.\) Tính tổng \(T=F\left( 0 \right)+F\left( 2 \right)+F\left( -\,3 \right).\)

Ta có \(f\left( x \right) = \left| {1 + x} \right| - \left| {1 - x} \right| = \left\{ \begin{array}{l}
2\,\,\,\,\,\,\,\,\,\,khi\,\,\,\,x \ge 1\\
2x\,\,\,\,\,\,\,\,khi\,\,\,\, - \,1 \le x < 1\\
- \,2\,\,\,\,\,\,\,khi\,\,\,\,\,x < - \,1
\end{array} \right. \Rightarrow F\left( x \right) = \left\{ \begin{array}{l}
2x + {C_1}\,\,\,\,\,\,\,\,khi\,\,\,\,x \ge 1\\
{x^2} + {C_2}\,\,\,\,\,\,\,\,khi\,\,\,\, - \,1 \le x < 1\\
- \,2x + {C_3}\,\,\,\,khi\,\,\,\,\,x < - \,1
\end{array} \right.\)

Theo đề bài ta có \(\left\{ \begin{array}{l}
F\left( 1 \right) = 3\\
F\left( { - 1} \right) = 2\\
F\left( { - 2} \right) = 4
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
2 + {C_1} = 3\\
1 + {C_2} = 2\\
4 + {C_3} = 4
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
{C_1} = 1\\
{C_2} = 1\\
{C_3} = 0
\end{array} \right.\)

\(\begin{array}{l}
\Rightarrow F\left( x \right) = \left\{ \begin{array}{l}
2x + 1\,\,\,\,\,\,\,\,khi\,\,\,\,x \ge 1\\
{x^2} + 1\,\,\,\,\,\,\,\,khi\,\,\,\, - \,1 \le x < 1\\
- \,2x\,\,\,\,khi\,\,\,\,\,x < - \,1
\end{array} \right. \Rightarrow \left\{ \begin{array}{l}
F\left( 2 \right) = 2.2 + 1 = 5\\
F\left( 0 \right) = 1\\
F\left( { - 3} \right) = - 2.\left( { - 3} \right) = 6.
\end{array} \right.\\
\Rightarrow T = 5 + 1 + 6 = 12.
\end{array}\)