Get the answer to "Determine whether the following integrals are convergen: ∫5∞arctan⁡xx2+3x+5dx"

fortdefruitI

Answered question

2021-05-18

Determine whether the following integrals are convergen:

\int_{5}^{\infty} \frac{\arctan x}{x^{2} + 3 x + 5} d x

coffentw

Skilled2021-05-19Added 103 answers

Solution:

\int_{5}^{\infty} \frac{\arctan x}{x^{2} + 3 x + 5} d x

Let

a_{n}

and

b_{n}

be two sequence such that far n,

a_{n} \leq b_{n}

- \frac{π}{2} \leq \arctan (x) \leq \frac{π}{2}

\Rightarrow \int_{5}^{\infty} \frac{\arctan n}{n^{2} + 3 n + 5} \leq \frac{\frac{π}{2}}{n^{2} + 3 n + 5}

Now,

a_{n} = \int_{5}^{\infty} \frac{1}{n^{2} + 3 n + 5}

a_{n} = \int_{5}^{\infty} \frac{1}{n^{2}}

a_{n} = \int_{5}^{\infty} \frac{1}{n^{2}}

- Converges

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