SATHEE: Indefinite Integration 3 Question 16

Indefinite Integration 3 Question 16

16. Which of the following is true?

(a) $g$ is increasing on $(1, \infty)$

(b) $g$ is decreasing on $(1, \infty)$

(d) $g$ is decreasing on $(1, 2)$ and increasing on $(2, \infty)$

Fill in the Blank

Show Answer

Answer:

Correct Answer: 16. (b)

Solution:

Here, $f (x) = (1 - x)^{2} \cdot \sin^{2} x + x^{2} \geq 0, \forall x$ .

and $g (x) = \int_{1}^{x} \frac{2 (t - 1)}{t + 1} - \log t f (t) d t$

$\Rightarrow g^{'} (x) = \frac{2 (x - 1)}{(x + 1)} - \log x \cdot \underset{+ v e}{\underset{⏟}{f (x)}}$

For $g^{'} (x)$ to be increasing or decreasing,

let $φ (x) = \frac{2 (x - 1)}{(x + 1)} - \log x$

$φ^{'} (x) = \frac{4}{(x + 1)^{2}} - \frac{1}{x} = \frac{- (x - 1)^{2}}{x (x + 1)^{2}}$

$φ^{'} (x) < 0$ , for $x > 1 \Rightarrow φ (x) < φ (1) \Rightarrow φ (x) < 0$

From Eqs. (i) and (ii), we get

$g^{'} (x) < 0 for x \in (1, \infty)$

$∴ g (x)$ is decreasing for $x \in (1, \infty)$ .

Indefinite Integration 3 Question 16

16. Which of the following is true?

Fill in the Blank

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Indefinite Integration 3 Question 16

16. Which of the following is true?

Fill in the Blank

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu