I want to figure out if: $$\int_0^1{\frac{1}{\sqrt{x+x^5}}}dx$$ is converging or diverging by using comparison. I cannot however figure out with what to compare it, the only integrand I can think of is the bigger one: $\frac{1}{\sqrt{x^5}}=\frac{1}{x^{5/2}}$ and I know since 5/2 > 1 that it diverges which doesn't help.
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Close to $0$, $x^5$ is neglectible in front of $x$. – Oct 25 '17 at 12:05
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$\int_{0}^{1}\frac{dx}{\sqrt{x}}$ is finite, so... – Jack D'Aurizio Oct 25 '17 at 15:48
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$$\frac{1}{\sqrt{x+x^5}} = \frac{1}{\sqrt{x}}\cdot \frac{1}{\sqrt{1+x^4}}$$
Near zero, the second factor is close to $1$, so you can "ignore" it, or you can just say it's smaller than $1$ to get the estimate you need.
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