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

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, ${a}_{n}\le {b}_{n}$
$-\frac{\pi }{2}\le \mathrm{arctan}\left(x\right)\le \frac{\pi }{2}$
$⇒{\int }_{5}^{\mathrm{\infty }}\frac{\mathrm{arctan}n}{{n}^{2}+3n+5}\le \frac{\frac{\pi }{2}}{{n}^{2}+3n+5}$
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

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