need assitance if possible 5. Use the comparison theorem to determine if the following integral converges.Clearly state the integral that you are comparing it to and show that this integralconverges.12x + 3x +41

Answered: 5. Use the comparison theorem to…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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need assitance if possible

5. Use the comparison theorem to determine if the following integral converges.
Clearly state the integral that you are comparing it to and show that this integral
converges.
1
2x + 3x +4
1

Expert Solution

Step 1

The objective here is to use comparison theorem, to show that the following integral converges.
$\int_{1}^{\infty} \frac{1}{2 x^{2} + 3 x + 4} d x$ .

The comparison test states the following:

If $f (x) \geq g (x) \geq 0,$ on the interval $[a, \infty)$ then,

a) If $\int_{a}^{\infty} f (x) d x$ converges, then so does $\int_{a}^{\infty} g (x) d x$ .
b) If $\int_{a}^{\infty} g (x) d x$ diverges, then so does $\int_{a}^{\infty} f (x) d x$ .

Step 2

Here, we have $f (x) = \frac{1}{2 x^{2} + 3 x + 4}$ .
Consider $g (x) = \frac{1}{2 x^{2}}$ .
As, $2 x^{2} + 3 x + 4 ⩾ 2 x^{2}, \forall x \in [1, \infty)$ ,
$\frac{1}{2 x^{2}} ⩾ \frac{1}{2 x^{2} + 3 x + 4}, \forall x \in [1, \infty)$
$\Rightarrow g (x) ⩾ f (x), \forall x \in [1, \infty)$

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