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Determine whether the integral is convergent or divergent. If it is convergent, ...

Jan 26, 2024

Determine whether the integral is convergent or divergent. If it is convergent, evaluate it.

\int_{-\infty}^{\infty} 68 \frac{x^{2}}{9+x^{6}} d x

Solution by Steps

step 1

To determine if the integral is convergent or divergent, we need to consider the behavior of the function as

x

approaches

\pm \infty

step 2

The function

\frac{68x^2}{9+x^6}

behaves like

\frac{1}{x^4}

x

approaches

\pm \infty

, which means the integral converges since the integral of

\frac{1}{x^4}

from

-\infty

\infty

is convergent

step 3

The result from the Asksia-LL calculator shows that the integral evaluates to

\frac{68\pi}{9}

Answer

\frac{68\pi}{9}

Key Concept

Convergence of Improper Integrals

Explanation

An improper integral is convergent if the limit of the integral as

x

approaches the bounds of integration exists and is finite. In this case, the function

\frac{68x^2}{9+x^6}

decreases rapidly enough as

x

approaches

\pm \infty

for the integral to converge.