Evaluate (int_{-1}^{0} (x+1)(x-2) mathrm{d}x)
-
A.
(frac{7}{6}) -
B.
(frac{5}{6}) -
C.
(frac{-5}{6}) -
D.
(frac{-7}{6})
Correct Answer: Option D
Explanation
Expanding ((x+1)(x-2) = x^{2} – 2x + x – 2 = x^{2} – x – 2)
(int_{-1}^{0} (x^{2} – x – 2) mathrm{d}x = [frac{x^{3}}{3} – frac{x^{2}}{2} – 2x]_{-1}^{0})
= ([frac{0}{3} – frac{0}{2} – 2times0 – (frac{-1^{3}}{3} – frac{-1^{2}}{2} – 2times-1)])
= (0 + frac{1}{3} + frac{1}{2} – 2 = frac{-7}{6})
Note: This can also be solved using integration by parts.
(int uv mathrm{d}x = uint v mathrm{d}x – int u'(int v mathrm{d}x)mathrm{d}x).