How to determine convergence of $\int _0^1 \frac{3cosx}{x^2+x^\frac{1}{2}}dx$

Question

I'm stuck on how to determine if the improper integral $$\int _0^1 \frac{3\cos x}{x^2+x^\frac{1}{2}}dx$$ converges.

The only tool I have (I think) is to apply the fundamental theorem of calculus and try to find the antiderivative, but I'm convinced its not feasible to find the antiderivative

Any pointers or hints would be appreciated!

Answer 1

Let $f(x)=3\cos(x)(x^2+x^{1/2})^{-1}$. Since $f$ is continuos at $x=1$, we have a unique problem in $x=0$. We can use de comparation criteria by taking limit. Since $$ \lim_{x\to 0} \frac{f(x)}{1/\sqrt{x}} = 1 $$ and $$ \int_{0}^1\frac{1}{\sqrt{x}}dx = 2< \infty $$ We have the result. The improper integral is convergent.