I am attempting to demonstrate the convergence of the integral $\int_{0}^{1} \frac{1}{1+x^{a+2}} \, dx$. My approach involves showing that for every positive value of $a$, the integral is definite and can be solved using partial fractions. However, I'm uncertain about the correctness of my approach.
If anyone has insights into whether this method is valid or if there are alternative approaches to establish the convergence of this integral, I would greatly appreciate any assistance or guidance. Thank you!
The only problem you could have would be at $x = 0$. Let's see
If $a > -2$ $$ \lim_{x \to 0^{+}} \frac{1}{1+x^{2+a}} = 1 $$ If $a = -2$ $$ \lim_{x \to 0^{+}} \frac{1}{1+x^{2+-2}} = 1/2 $$ If $a < -2$ $$ \lim_{x \to 0^{+}} \frac{1}{1+x^{2+a}} = 0 $$ Since the improper integral is of type 2, it converges. So the integral converges for any $a \in \mathbb{R}$