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Question
Evaluate the following integral:
Solution
\[\int_{- 2}^2 \left| 2x + 3 \right| d x\]
\[We\ know\ that\ \left| 2x + 3 \right| = \begin{cases} - \left( 2x + 3 \right) &, &- 2 \leq x \leq - \frac{3}{2}\\\left( 2x + 3 \right)&, &- \frac{3}{2} < x \leq 2\end{cases}\]
\[ \therefore I = \int_{- 2}^\frac{- 3}{2} - \left( 2x + 3 \right) d x + \int_{- \frac{3}{2}}^2 \left( 2x + 3 \right) d x\]
\[ \Rightarrow I = - \left[ x^2 + 3x \right]_{- 2}^\frac{- 3}{2} + \left[ x^2 + 3x \right]_{- \frac{3}{2}}^2 \]
\[ \Rightarrow I = - \frac{9}{4} + \frac{9}{2} + 4 - 6 + 4 + 6 - \frac{9}{4} + \frac{9}{2}\]
\[ \Rightarrow I = \frac{25}{2}\]
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