Hi, I don't know to determine the convergence of these integrals:

The first one is $\displaystyle {\displaystyle \int_{0}^{1}\frac{3\sin^{2}(x)+5\cos^{5}(x)}{\sqrt {x}(x+1)}}dx $

and the second is $\displaystyle {\displaystyle \int_{1}^{\infty}\frac{2+x^{3}}{1+x^{6}}dx}$

$\displaystyle \displaystyle$

For the second I tried the following $\displaystyle

{\displaystyle I=\int_{1}^{\infty}\frac{2}{x^{6}(1+\frac{1}{x^{6} })}dx}+\int_{1}^{\infty}\frac{1}{x^{3}(1+\frac{1}{ x^{6}})}dx=\int_{1}^{\infty}f_{1}(x)dx+\int_{1}^{\ infty}f_{2}(x)dx

$

When $\displaystyle \displaystyle x\rightarrow\infty \begin{cases}

x^{6}\rightarrow & \infty\\

(1+\frac{1}{x^{6}}) & \rightarrow1\\

x^{3} & \rightarrow\infty\end{cases}$

Then

$\displaystyle f_{1}(x)\sim\frac{2}{x^{6}}$ , Thus$\displaystyle I_{1}$converges

$\displaystyle {\displaystyle f_{2}(x)}\sim\frac{1}{x_{3}}\Rightarrow$ $\displaystyle I_{2}$ converges, then$\displaystyle I$ converges.

my reasoning is correct?

Bye and thanks for everything.