Hello i have a question pplz help me it is
how the:
a is a constant number
after integration became
I really appreciate your help
Thanks
This problem is very drawn out for me, I will find out how to do it though! It will be good for both of us, but our methods seem to complicate thing while integrating and simplifying as I told by my professor, mathematicians aim to make hard things very simple when working out a problem, so we will get it! I am sure there a few around here that this is easy for, I was close to the solution but made a careless everyone and wound up with a few missing parts, like the numerator coming to p^2 instead of p^3
Lets see if we can make sense of things together, first we know -3a is a constant so we can pull that to the outside and that the numerator $\displaystyle p^2$ when integrated as if it was just alone as such$\displaystyle p^2$ becomes $\displaystyle \frac{p^3}{3}$ and remember we pulled out the -3a so now the the integrated form we get $\displaystyle -3a*\frac{p^3}{3}$ we get $\displaystyle -a * p^3$ so somehow we need to make a relation to the denominator and decide the right method of integrating to where we can relate to this
Edit: have you tried integration by parts, I feel I am close using this method
Have you tried the substitution $\displaystyle p = \frac{1}{x} $ ?
since
$\displaystyle dp = \frac{-1}{x^2} dx $
the integral ( let's omit the constant $\displaystyle -3a $) becomes
$\displaystyle \int \frac{x^5}{\sqrt{x^2+1}^5} \cdot \frac{1}{x^2} \cdot \frac{-1}{x^2} dx $
$\displaystyle = - \int \frac{x dx}{\sqrt{x^2+1}^5} $
$\displaystyle = - (1 + x^2)^{\frac{-3}{2}} (\frac{-2}{3}) \frac{1}{2} + C $
$\displaystyle = \frac{1}{3} \frac{ 1}{ \sqrt{ 1 + \frac{1}{p^2} }^3 } + C$
$\displaystyle = \frac{ p^3}{3 \sqrt{ 1 + p^2}^3 } + C $
If thec0o0lest didn't give us the answer , i won't think of this substitution .
I observed that the integral is algebraic function , not transcendental ( usually $\displaystyle \sinh^{-1}(x) , \cosh^{-1}(x) $ ) .
so i attempt to change the variable ( algebraic ) . It is just a trial !
Another case we can use this sub. is
$\displaystyle \int \frac{dx}{\sqrt{x^2+1}^3} $
to
$\displaystyle - \int \frac{u ~du}{\sqrt{u^2+1}^3} $