Question

Evaluate the following integral:

\[\int\limits_{- 3}^3 \left| x + 1 \right| dx\]

Answer 1

\[I = \int_{- 3}^3 \left| x + 1 \right| d x\]
\[We\ know\ that, \left| x + 1 \right| = \begin{cases} - \left( x + 1 \right) &, &- 3 \leq x \leq - 1 \\x + 1 &, &- 1 < x \leq 3\end{cases}\]
\[ \therefore I = \int_{- 3}^{- 1} - \left( x + 1 \right) d x + \int_{- 1}^3 \left[ x + 1 \right] d x\]
\[ \Rightarrow I = \left[ - \frac{\left( x + 1 \right)^2}{2} \right]_{- 3}^{- 1} + \left[ \frac{\left( x + 1 \right)^2}{2} \right]_{- 1}^3 \]
\[ \Rightarrow I = 0 + 2 + 8 - 0\]
\[ \Rightarrow I = 10\]