Question

Evaluate the following integral:

\[\int\limits_{- 2}^2 \left| x + 1 \right| dx\]

 

Answer 1

\[\int_{- 2}^2 \left| x + 1 \right| d x\]
\[\text{We know that}, \left| x + 1 \right| = \begin{cases} - \left( x + 1 \right) &,& - 2 \leq x \leq - 1\\x + 1&,& - 1 < x \leq 2\end{cases}\]
\[ \therefore I = \int_{- 2}^2 \left| x + 1 \right| d x\]
\[ \Rightarrow I = \int_{- 2}^{- 1} - \left( x + 1 \right) dx + \int_{- 1}^2 \left( x + 1 \right) dx\]
\[ \Rightarrow I = \left[ \frac{- x^2}{2} - x \right]_{- 2}^{- 1} + \left[ \frac{x^2}{2} + x \right]_{- 1}^2 \]
\[ \Rightarrow I = \frac{- 1}{2} + 1 + 2 - 2 + 2 + 2 - \frac{1}{2} + 1\]
\[ \Rightarrow I = 5\]