$Convergence of integrals$

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

Calculus Improper Integrals Convergence

Phia

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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

Phia

Why did you put 2 in numerator?
Savionf

See the revised solution.

$Convergence of integrals$

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