SATHEE: Indefinite Integration 1 Question 42

Indefinite Integration 1 Question 42

43. If $\int_{1}^{3} x^{2} F^{'} (x) d x = - 12$ and $\int_{1}^{3} x^{3} F^{''} (x) d x = 40$ , then the correct expression(s) is/are

(a) $9 f^{'} (3) + f^{'} (1) - 32 = 0$

(b) $\int_{1}^{3} f (x) d x = 12$

(d) $\int_{1}^{3} f (x) d x = - 12$

Passage II

For every function $f (x)$ which is twice differentiable, these will be good approximation of

$\int_{a}^{b} f (x) d x = \frac{b - a}{2} f (a) + f (b),$

for more acurate results for $c \in (a, b)$ ,

$F (c) = \frac{c - a}{2} [f (a) - f (c)] + \frac{b - c}{2} [f (b) - f (c)]$

When $c = \frac{a + b}{2}$

$\int_{a}^{b} f (x) d x = \frac{b - a}{4} f (a) + f (b) + 2 f (c) d x$

$(2006, 6 M)$

Show Answer

Solution:

Given, $\int_{1}^{3} x^{2} F^{'} (x) d x = - 12$

$\Rightarrow {[x^{2} F (x)]}_{1}^{3} - \int_{1}^{3} 2 x \cdot F (x) d x = - 12$

$\Rightarrow 9 F (3) - F (1) - 2 \int_{1}^{3} f (x) d x = - 12$

$\begin{aligned} \Rightarrow - 36 - 0 - 2 \int_{1}^{3} f (x) d x = - 12 \\ ∴ \int_{1}^{3} f (x) d x = - 12 and \int_{1}^{3} x^{3} F^{''} (x) d x = 40 \end{aligned}$

$\Rightarrow {[x^{3} F^{'} (x)]}_{1}^{3} - \int_{1}^{3} 3 x^{2} F^{'} (x) d x = 40$

$\Rightarrow [x^{2} {(x F^{'} (x)]}_{1}^{3} - 3 \times (- 12) = 40$

$\Rightarrow {x^{2} \cdot [f^{'} (x) - F (x)]}_{1}^{3} = 4$

$\Rightarrow 9 [f^{'} (3) - F (3)] - [f^{'} (1) - F (1)] = 4$

$\Rightarrow 9 [f^{'} (3) + 4] - [f^{'} (1) - 0] = 4$

$\Rightarrow 9 f^{'} (3) - f^{'} (1) = - 32$

Indefinite Integration 1 Question 42

43. If $\int_{1}^{3} x^{2} F^{'} (x) d x = - 12$ and $\int_{1}^{3} x^{3} F^{''} (x) d x = 40$ , then the correct expression(s) is/are

Passage II

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 42

43. If ∫13x2F′(x)dx=−12 and ∫13x3F′′(x)dx=40, then the correct expression(s) is/are

Passage II

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

43. If $\int_{1}^{3} x^{2} F^{'} (x) d x = - 12$ and $\int_{1}^{3} x^{3} F^{''} (x) d x = 40$ , then the correct expression(s) is/are