$Convergence of integrals$

Question

$Convergence of integrals$

$\int_{1}^{\infty}\frac{cos(e^x)+sen(e^{-x})}{e^{x}+e^{-x}}dx $

Calculus Improper Integrals Convergence

Phia

19

Report

Answer

The answer is accepted.

Join Matchmaticians Affiliate Marketing Program to earn up to a 50% commission on every question that your affiliated users ask or answer.

Answer 1

Answers can only be viewed under the following conditions:

The questioner was satisfied with and accepted the answer, or
The answer was evaluated as being 100% correct by the judge.

View the answer

We have
\[| \frac{\cos(e^x)+\sin(e^{-x})}{e^{x}+e^{-x}}dx|\leq \frac{|\cos(e^x)|+|\sin(e^{-x})|}{e^{x}+e^{-x}}dx\]
\[\leq \frac{2}{e^{x}+e^{-x}}dx \leq \frac{2}{e^x}, \]
because $|\cos(e^x)| \leq 1$ and $|\sin(e^x)|\leq 1$.

Since $\int_{1}^{\infty}\frac{2}{e^x}dx$ is convergenct, by comparison theorem the given integral is also absolutely convergent, and hence convergent.

Savionf

574

Phia

0

Why did you put 2 in numerator?
Savionf

0

See the revised solution.

$Convergence of integrals$

Answer

Related Questions

Search