SATHEE: Definite Integration Question 10

Definite Integration Question 10

Question 10

Let $f : (0, \infty) \to R$ and $F (x) = \int_{0}^{x} f (t) d t$ .

If $F (x^{2}) = x^{2} (1 + x)$ , then $f (4)$ equals (a) $\frac{5}{4}$ (b) 7 (c) 4 (d) 2

Show Answer

Answer:

Correct Answer: 10. (c)

Solution:

Given, $F (x) = \int_{0}^{x} f (t) d t$

By Leibnitz’s rule,

$F^{'} (x) = f (x)$

But $F (x^{2}) = x^{2} (1 + x) = x^{2} + x^{3}$

[given]

$\begin{array}{ll} \Rightarrow & F (x) = x + x^{3 / 2} \Rightarrow F^{'} (x) = 1 + \frac{3}{2} x^{1 / 2} \Rightarrow & f (x) = F^{'} (x) = 1 + \frac{3}{2} x^{1 / 2} [from Eq. (i)] \Rightarrow & f (4) = 1 + \frac{3}{2} (4)^{1 / 2} \Rightarrow f (4) = 1 + \frac{3}{2} \times 2 = 4 \end{array}$

Definite Integration Question 10

Question 10

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Definite Integration Question 10

Question 10

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu