Expanding (x+1)(x−2)=x2−2x+x−2=x2−x−2(x+1)(x-2) = x^{2} - 2x + x - 2 = x^{2} - x - 2(x+1)(x−2)=x2−2x+x−2=x2−x−2
∫−10(x2−x−2)dx=[ x33− x22−2x]−10\int\limits_{-1}^{0} (x^{2} - x - 2) dx = \left[ \;\frac{x^{3}}{3} - \;\frac{x^{2}}{2} - 2x \right]_{-1}^{0}−1∫0(x2−x−2)dx=[3x3−2x2−2x]−10
= [ 03− 02−2×0−( −133− −122−2×−1)]\left[ \;\frac{0}{3} - \;\frac{0}{2} - 2 \times 0 - \left( \;\frac{-1^{3}}{3} - \;\frac{-1^{2}}{2} - 2 \times -1 \right) \right][30−20−2×0−(3−13−2−12−2×−1)]
= 0+ 13+ 12−2= −760 + \;\frac{1}{3} + \;\frac{1}{2} - 2 = \;\frac{-7}{6}0+31+21−2=6−7
Note: This can also be solved using integration by parts.
∫uv dx=u∫v dx−∫u′(∫v dx)dx\int u v \, dx = u \int v \, dx - \int u' \left( \int v \, dx \right) dx∫uvdx=u∫vdx−∫u′(∫vdx)dx.