The value of $\int\limits_{-5}^5|x+3|dx$ is

34

The correct answer is Option (3) → 34

Integral: $\int_{-5}^{5} |x+3| \, dx$

Split at $x = -3$ where $x+3 = 0$:

$\int_{-5}^{-3} |x+3| \, dx + \int_{-3}^{5} |x+3| \, dx$

For $x \in [-5, -3]$, $x+3 \le 0 \Rightarrow |x+3| = -(x+3)$

$\int_{-5}^{-3} -(x+3) \, dx = \int_{-5}^{-3} (-x - 3) \, dx = \left[ -\frac{x^2}{2} - 3x \right]_{-5}^{-3}$

$= \left(-\frac{9}{2} + 9 \right) - \left(-\frac{25}{2} + 15\right) = \frac{9}{2} - (-\frac{25}{2} +15) = \frac{9}{2} - (-\frac{25}{2} +15) = \frac{9}{2} + \frac{25}{2} -15 = 17 -15 = 2$

For $x \in [-3,5]$, $x+3 \ge 0 \Rightarrow |x+3| = x+3$

$\int_{-3}^{5} (x+3) \, dx = \left[ \frac{x^2}{2} + 3x \right]_{-3}^{5} = \left(\frac{25}{2} + 15\right) - \left(\frac{9}{2} -9\right) = \frac{25}{2}+15 - (\frac{9}{2}-9) = \frac{25-9}{2}+15+9 = 8+24 =32$

Total integral: $2 + 32 = 34$