Determine whether the following integrals are convergen: ∫5∞arctanxx2+3x+5dx

fortdefruitI

Answered question

2021-05-18

Determine whether the following integrals are convergen:
${\int}_{5}^{\mathrm{\infty}}\frac{\mathrm{arctan}x}{{x}^{2}+3x+5}dx$

Answer & Explanation

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