First slide

Methods of integration

Question

Let $f : (11, \infty) \to (0, \infty)$ be given by $f (x) = \prod_{l = 1}^{10} \frac{1}{(x - l)}$
where for real number $a_{1} . . . . . . . . . . a_{n}$
$\prod_{l = 1}^{n} a_{l}$ denotes the product $a_{1} \times a_{2} \dots \times a_{n}$
Statement 1: $\int f (x) d x = \sum_{l = 1}^{10} \frac{(- 1)^{l} \log | x - l |}{(l - 1)! (10 - l)!}$
Statement 2: For $x \in [11, \infty) f (x) = \sum_{l = 1}^{10} \frac{A_{l}}{x - l}$ where $A_{1} = \prod_{j = 1}^{10} \frac{j}{l - j} l = 1, 2, 10$

Difficult

A

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
B

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
C

STATEMENT-1 is True, STATEMENT-2 is False
D

STATEMENT-1 is False, STATEMENT-2 is True

Solution

$f (x) = \prod_{l = 1}^{10} \frac{1}{x - l} = \frac{A_{1}}{x - 1} + \frac{A_{2}}{x - 2} + \dots + \frac{A_{10}}{x - 10}$
where $A_{i} = \frac{1}{(i - 1) \dots (i - (i + 1)) \dots (i - 10)}$
$= \frac{(- 1)^{i}}{(i - 1)! (10 - i)!}$
$\int f (x) d x = \sum_{i = 1}^{10} \int \frac{A_{i}}{x - i} d x = \sum_{i = 1}^{10} A_{i} \log | x - i |$
So statement $1$ is true but not statement $2$ .

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App