If $[t]$ denotes the greatest integer $\leq t$, then the value of $\int_{0}^{1}\left[2\mathrm{x}-\left|3\mathrm{x}^{2}-5\mathrm{x}+2\right|+1\right]\mathrm{dx}$ is:

(A) $\frac{\sqrt{37}+\sqrt{13}-4}{6}$

(B) $\frac{\sqrt{37}-\sqrt{13}-4}{6}$

(C) $\frac{-\sqrt{37}-\sqrt{13}+4}{6}$

(D) $\frac{-\sqrt{37}+\sqrt{13}+4}{6}$

JEE Main · 2022 · 29 Jul 2022 · Shift 2 · Definite Integrals

[t]

denotes the greatest integer

\leq t

, then the value of

\int_{0}^{1} [2 x - 3 x^{2} - 5 x + 2 + 1] dx

is:

Answer: (A)

First, factor

3 x^{2} - 5 x + 2 = (3 x - 2) (x - 1)

. The expression changes sign at

x = 2/3

and

x = 1

. Split the integral:

\int_{0}^{2/3} [- 3 x^{2} + 7 x - 2] d x + \int_{2/3}^{1} [3 x^{2} - 3 x + 2] d x + 1

. For the first part, evaluate the greatest integer function by finding where

- 3 x^{2} + 7 x - 2

crosses integer values. For the second part,

3 x^{2} - 3 x + 2 \in (4/3, 2)

(2/3, 1)

, so

[3 x^{2} - 3 x + 2] = 1

. After careful evaluation of the roots and integration, the result is

\frac{37 + 13 - 4}{6}

JEE Main · 2022 · 29 Jul 2022 · Shift 2 · Single correct option