SATHEE: Indefinite Integration 1 Question 31

Indefinite Integration 1 Question 31

32. Let $f (x) = x - [x]$ , for every real number $x$ , where $[x]$ is the integral part of $x$ . Then, $\int_{- 1}^{1} f (x) d x$ is

(1998, 2M)

(a) 1

(b) 2

(d) $- \frac{1}{2}$

Show Answer

Solution:

Let $\int_{- 1}^{1} f (x) d x = \int_{- 1}^{1} (x - [x]) d x = \int_{- 1}^{1} x d x - \int_{- 1}^{1} [x] d x$ $= 0 - \int_{- 1}^{1} [x] d x [∵ x$ is an odd function $]$

$- 1, if - 1 \leq x < 0$

But

$[x] = 0$ , if $0 \leq x < 1$

$1, if x = 1$

$∴ \int_{- 1}^{1} [x] d x = \int_{- 1}^{0} [x] d x + \int_{0}^{1} [x] d x$ $= \int_{- 1}^{0} (- 1) d x + \int_{0}^{1} 0 d x$ $= - [x]_{- 1}^{0} + 0 = - 1; ∴ \int_{- 1}^{1} f (x) d x = 1$

Indefinite Integration 1 Question 31

32. Let $f (x) = x - [x]$ , for every real number $x$ , where $[x]$ is the integral part of $x$ . Then, $\int_{- 1}^{1} f (x) d x$ is

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Indefinite Integration 1 Question 31

32. Let f(x)=x−[x], for every real number x, where [x] is the integral part of x. Then, ∫−11f(x)dx is

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

32. Let $f (x) = x - [x]$ , for every real number $x$ , where $[x]$ is the integral part of $x$ . Then, $\int_{- 1}^{1} f (x) d x$ is