I need to check if I am right solving this integral

[mcastillo356]

TL;DR Summary

I have a list of elementary integrals, and among them one that involves arctangent; the example I am dealing with is a combination I will propose in the next discussion paragraph.

Hi, PF

1-The elementary integral is $\displaystyle\int{\displaystyle\frac{1}{a^2+x^2}dx}=\displaystyle\frac{1}{a}\tan^{-1}\displaystyle\frac{x}{a}+C$

2-The example is $\displaystyle\int{\Big(5x^{3/5}-\displaystyle\frac{3}{2+x^2}\Big)dx}=\displaystyle\frac{25}{8}x^{8/5}-\displaystyle\frac{3}{\sqrt{2}}\tan^{-1}\displaystyle\frac{x}{\sqrt{2}}+C$

The question is: does the first statement agree with the solution showed?; any comment?

Greetings!

PD: I post without preview.

 

[anuttarasammyak]

Yes, it does. What is your uneasiness ?

 

[mcastillo356]

Yes, it does. What is your uneasiness ?

Thanks a lot! I needed some help. I confess maths are among my favorite fields, but I am not specially good at them. I was quite sure, but still wanted to share with somebody. It was just some kind of necessity to put things in common; just ease my loneliness at this ground so interesting to me.
P&L.
Greetings!