Define a function $f: R \rightarrow R$ by $f(x)=\max \{|x|,|x-1|, \ldots \ldots .|x-2 n|\}$ Where $\mathrm{n}$ is a fixed natural number, Then $\int_{0}^{2 n} f(x) d x$ is

• A

  $\mathrm{n}$
• B

  $\mathrm{n}^{2}$
• C

  $3 \mathrm{n}$
• Correct answer
  D

  $3 \mathrm{n}^{2}$

Solution

$\int_{0}^{2 n} f(x) d x=\int_{0}^{n}|x-2 n| d x+\int_{n}^{2 n}|x| d x$ $\int_{0}^{2 n}(2 n-x) d x+\int_{n}^{2 n} x d x$ $=\left(2 n x-\frac{x^{2}}{2}\right)_{0}^{n}+\left(\frac{x^{2}}{2}\right)_{n}^{2 n}=3 n^{2}$

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