Question

Evaluate the following integral:

\[\int\limits_{- 6}^6 \left| x + 2 \right| dx\]

 

Answer 1

\[\int_{- 6}^6 \left| x + 2 \right| d x\]
\[\text{We know that}, \left| x + 2 \right| = \begin{cases} - \left( x + 2 \right) &,& - 6 \leq x \leq - 2\\x + 2&,& - 2 < x \leq 6\end{cases}\]
\[ \therefore I = \int_{- 6}^6 \left| x + 2 \right| d x\]
\[ \Rightarrow I = \int_{- 6}^{- 2} - \left( x + 2 \right) dx + \int_{- 2}^6 \left( x + 2 \right) dx\]
\[ \Rightarrow I = \left[ \frac{- x^2}{2} - 2x \right]_{- 6}^{- 2} + \left[ \frac{x^2}{2} + 2x \right]_{- 2}^6 \]
\[ \Rightarrow I = - 2 + 4 + 18 - 12 + 18 + 12 - 2 + 4\]
\[ \Rightarrow I = 40\]