Question

\[ \int_{-2}^2\left|x^2-x-2\right| \mathrm{d} x= \]

• A. \(\frac{17}{3}\)
• B. \(\frac{19}{3}\)
• C. 19
• D. 17

Hint:

Find where x^2 - x - 2 changes sign and split the integral at those points.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Factorise \(x^{2}-x-2=(x-2)(x+1)\). It is positive for \(x \lt -1\) and negative for \(-1 \lt x \lt 2\).

Step 2: Split the integral
\(\int_{-2}^{2}|x^{2}-x-2|dx=\int_{-2}^{-1}(x^{2}-x-2)dx-\int_{-1}^{2}(x^{2}-x-2)dx\).

Step 3: Evaluate
With \(F(x)=\dfrac{x^{3}}{3}-\dfrac{x^{2}}{2}-2x\): \(F(-2)=-\dfrac{2}{3}\), \(F(-1)=\dfrac{7}{6}\), \(F(2)=-\dfrac{10}{3}\).
First part: \(F(-1)-F(-2)=\dfrac{11}{6}\). Second part: \(-[F(2)-F(-1)]=\dfrac{9}{2}\).

Step 4: Add
Total \(=\dfrac{11}{6}+\dfrac{27}{6}=\dfrac{19}{3}\).

Final Answer:
The value of the integral is 19/3. \[ \boxed{\text{(B) }\dfrac{19}{3}} \]