Definite Integration Question 42

Question 42

  1. If 13x2F(x)dx=12 and 13x3F(x)dx=40, then the correct expression(s) is/are (a) 9f(3)+f(1)32=0 (b) 13f(x)dx=12 (c) 9f(3)f(1)+32=0 (d) 13f(x)dx=12

Passage II

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

abf(x)dx=ba2f(a)+f(b),

for more acurate results for c(a,b),

F(c)=ca2[f(a)f(c)]+bc2[f(b)f(c)]

When c=a+b2

abf(x)dx=ba4f(a)+f(b)+2f(c)dx

(2006,6M)

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Solution:

  1. Given, 13x2F(x)dx=12

$\Rightarrow \quad\left[x^{2} F(x)\right]{1}^{3}-\int{1}^{3} 2 x \cdot F(x) d x=-12$

9F(3)F(1)213f(x)dx=12

360213f(x)dx=12 13f(x)dx=12 and 13x3F(x)dx=40

$\Rightarrow \quad\left[x^{3} F^{\prime}(x)\right]{1}^{3}-\int{1}^{3} 3 x^{2} F^{\prime}(x) d x=40$

[x2(xF(x)]133×(12)=40

Missing or unrecognized delimiter for \left

9[f(3)F(3)][f(1)F(1)]=4

9[f(3)+4][f(1)0]=4

9f(3)f(1)=32



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