Use the comparison test to determine whether the integral below converges or diverges.Make sure any inequality you write is very well justified.e" (sin² x)dre2x + 3+ x2

Note sin2x≤1 so exsin2xe2x+3+x2≤exe2x+3+x2.

Answered: Use the comparison test to determine…

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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Question

Use the comparison test to determine whether the integral below converges or diverges.
Make sure any inequality you write is very well justified.
e" (sin² x)
dr
e2x + 3+ x2

Expert Solution

Step 1

Note $\sin^{2} x \leq 1$ so $\frac{e^{x} (si n^{2} x)}{e^{2 x} + 3 + x^{2}} \leq \frac{e^{x}}{e^{2 x} + 3 + x^{2}}$ .

Step 2

Also, $3 + x^{2} > 0$ so we have:

$3 + x^{2} > 0 \frac{e^{2 x} + 3 + x^{2}}{e^{x}} > \frac{e^{2 x}}{e^{x}} \frac{e^{x}}{e^{2 x} + 3 + x^{2}} < \frac{e^{x}}{e^{2 x}}$

Step 3

But since $\frac{e^{x} (si n^{2} x)}{e^{2 x} + 3 + x^{2}} \leq \frac{e^{x}}{e^{2 x} + 3 + x^{2}}$ so $\frac{e^{x} (si n^{2} x)}{e^{2 x} + 3 + x^{2}} \leq \frac{e^{x}}{e^{2 x} + 3 + x^{2}} < \frac{e^{x}}{e^{2 x}}$ or $\frac{e^{x} (si n^{2} x)}{e^{2 x} + 3 + x^{2}} < \frac{e^{x}}{e^{2 x}} = e^{- x}$ .

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