Let $[t]$ denote the largest integer less than or equal to $t$. If $\int_{0}^{3}\left(\left[x^{2}\right]+\left[\frac{x^{2}}{2}\right]\right) d x=a+b \sqrt{2}-\sqrt{3}-\sqrt{5}+c \sqrt{6}-\sqrt{7}$ where $a, b, c \in \mathbb{Z}$, then $a+b+c$ is equal to ______.

JEE Main · 2024 · 06 Apr 2024 · Shift 2 · Definite Integrals

Let

[t]

denote the largest integer less than or equal to

t

.

If

\int_{0}^{3} ([x^{2}] + [\frac{x ^{2}}{2}]) d x = a + b 2 - 3 - 5 + c 6 - 7

where

a, b, c \in Z

, then

a + b + c

is equal to ______.

Answer: 23

Splitting

\int_{0}^{3} [x^{2}] d x

x = 1, 2, 3, 2, 5, 6, 7, 8

and

\int_{0}^{3} [x^{2} /2] d x

x = 2, 2, 6, 8

, and summing each as a step function, gives

\int_{0}^{3} [x^{2}] d x = 21 - 32 - 3 - 5 - 6 - 7, \int_{0}^{3} [\frac{x ^{2}}{2}] d x = 10 - 32 - 6

Adding: total

= 31 - 62 - 3 - 5 - 26 - 7

. Matching

a + b 2 - 3 - 5 + c 6 - 7

gives

a = 31, b = - 6, c = - 2

, so

a + b + c = 23

JEE Main · 2024 · 06 Apr 2024 · Shift 2 · Integer answer

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