SATHEE: Indefinite Integration 3 Question 10

Indefinite Integration 3 Question 10

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

(d) 2

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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]

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

Indefinite Integration 3 Question 10

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

Answer:

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Indefinite Integration 3 Question 10

10. Let f:(0,∞)→R and F(x)=∫0xf(t)dt.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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