If [x] denotes the greatest integer less than | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let $I = \int_{1}^{5} [|x - 3|] \, dx$. We split the integral based on the definition of the absolute value and the greatest integer function: $I

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(B) Let $I = \int_{1}^{5} [|x - 3|] \, dx$. We split the integral based on the definition of the absolute value and the greatest integer function: $I = \int_{1}^{3} [-(x - 3)] \, dx + \int_{3}^{5} [x - 3] \, dx$. For the first part,$\int_{1}^{3} [3 - x] \, dx$: When $1 \le x When $2 \le x Thus,$\int_{1}^{3} [3 - x] \, dx = \int_{1}^{2} 1 \, dx + \int_{2}^{3} 0 \, dx = [x]_{1}^{2} = 2 - 1 = 1$. For the second part,$\int_{3}^{5} [x - 3] \, dx$: When $3 \le x When $4 \le x Thus,$\int_{3}^{5} [x - 3] \, dx = \int_{3}^{4} 0 \, dx + \int_{4}^{5} 1 \, dx = [x]_{4}^{5} = 5 - 4 = 1$. Therefore,$I = 1 + 1 = 2$.

If $[x]$ denotes the greatest integer less than or equal to $x$,then the value of $\int_{1}^{5} [|x - 3|] \, dx$ is

Similar Questions

$ \int_{0}^{2} [x^{2}] \, dx $

Let $[x]$ and $\{x\}$ be the integer part and fractional part of a real number $x$ respectively. The value of the integral $\int_0^5 [x]\{x\} dx$ is

$\int_2^3 \frac{\log x}{x} d x=$

The value of $\int_{0}^{1} |3x^2 - 1| dx$ is

$\int_{ - \pi /2}^{\pi /2} {\sqrt {\frac{1}{2}(1 - \cos 2x)} } \,dx = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module