fortdefruitI

2021-05-18

Determine whether the following integrals are convergen:

${\int}_{5}^{\mathrm{\infty}}\frac{\mathrm{arctan}x}{{x}^{2}+3x+5}dx$

coffentw

Skilled2021-05-19Added 103 answers

Solution:

${\int}_{5}^{\mathrm{\infty}}\frac{\mathrm{arctan}x}{{x}^{2}+3x+5}dx$

Let${a}_{n}$ and ${b}_{n}$ be two sequence such that far n, ${a}_{n}\le {b}_{n}$

$-\frac{\pi}{2}\le \mathrm{arctan}(x)\le \frac{\pi}{2}$

$\Rightarrow {\int}_{5}^{\mathrm{\infty}}\frac{\mathrm{arctan}n}{{n}^{2}+3n+5}\le \frac{\frac{\pi}{2}}{{n}^{2}+3n+5}$

Now,

${a}_{n}={\int}_{5}^{\mathrm{\infty}}\frac{1}{{n}^{2}+3n+5}$

${a}_{n}={\int}_{5}^{\mathrm{\infty}}\frac{1}{{n}^{2}}$

${a}_{n}={\int}_{5}^{\mathrm{\infty}}\frac{1}{{n}^{2}}$ - Converges

Let

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