[tex]\displaystyle\int_{-3}^5 f(t)\, dt = \int_{-3}^{-1}(t+3)\, dt +\int_{-1}^3 2\, dt +\int_{3}^5(-t+5)\, dt = \\ \\ = \Big(\dfrac{t^2}{2}+3t\Big)\Bigg|_{-3}^{-1}+2t\Big|_{-1}^3+\Big(-\dfrac{t^2}{2}+5t\Big)\Bigg|_{3}^5 = \\ \\\\=\dfrac{1}{2}-3+6-\dfrac{25}{2}+25 -\Big(\dfrac{9}{2}-9-2-\dfrac{9}{2}+15\Big) = \\ \\ = -\dfrac{24}{2}+24 = -12+24 = \boxed{12}[/tex]
