Q92 / 120 · NDA 2024 (Phase 2) · Mathematics

Paragraph: Let $f(x)=\left|x^2-x-2\right|$
Question: What is $\int_1^3 f(x) d x$ equal to?

  • A
    2
  • Correct answer
    B
    3
  • C
    4
  • D
    5

Solution

$\begin{aligned} & \text { } f(x)=\left|x^2-x-2\right| \\ & \qquad\left\{\begin{array}{l}x^2-x-2 ; x \in(-\infty,-1] \cup[2, \infty) \\ -\left(x^2-x-2\right) ; x \in[-1,2]\end{array}\right. \\ & \text {Let } I=\int_1^3 f(x) d n=-\int_1^2\left(x^2-x-2\right) d x+\int_2^3\left(x^2-x-2\right) d x \\ & \quad=-\left[\frac{x^3}{3}-\frac{x^2}{2}-2 x\right]_1^2+\left[\frac{x^3}{3}-\frac{x^2}{2}-2 x\right]_2^3 \\ & =-\left[\frac{8}{3}-\frac{4}{2}-4\right]+\left[\frac{1}{3}-\frac{1}{2}-2\right]+\left[\frac{27}{3}-\frac{9}{2}-6\right]-\left[\frac{8}{3}-\frac{4}{2}-4\right] \\ & \quad=\frac{20}{6}-\frac{13}{6}-\frac{9}{6}+\frac{20}{6}=\frac{18}{6}=3\end{aligned}$

Hence, the correct answer is option: 'B'.

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