Solutions - 0130-01

(1) Find comparable integrand:

Higher power dominates for large $x$:

$$\frac{1}{x^4+\sqrt{x}}⟶\frac{1}{x^4}\text{as }x\to \infty$$

Therefore, compare to $1/x^4$.

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(2) Make comparison:

$$\frac{1}{x^4+\sqrt{x}}<\frac{1}{x^4}\text{for all }x\ge 1$$

And:

$${\int }_1^{\infty }\frac{1}{x^4}dx\text{converges}$$

because it is a $p$-integral with $p=4>1$.

By the Comparison Test, we conclude that:

$${\int }_1^{\infty }\frac{1}{x^4+\sqrt{x}}dx\text{converges}$$