Euler Substitutions vs Trigonometric Substitutions

[Al-Layth]

Both the Euler subs and the Trigonometric Substitutions aim to Simplify integrands contains a radical quadratic polynomial:
[math]\sqrt{ ax^{2} +bx +c }[/math]
So which substitution is better for what type of integrand?
Also I heard euler substitutions are "more general" so can be used in more integrands. Can someone explain this pls

thanks

 

[Steven G]

You can use Euler substitution when you have an integrand in terms of x and sqrt(ax^2 + bx + c), that is in the form \(\displaystyle \int f(x,\sqrt{(ax^2+bx+c)}dx\)

 

[Al-Layth]

You can use Euler substitution when you have an integrand in terms of x and sqrt(ax^2 + bx + c), that is in the form \(\displaystyle \int f(x,\sqrt{(ax^2+bx+c)}dx\)

ive seen trig sub used when the integrand is a function of x and the radical quadratic too
can u give me an example integrand where only one of these substitutions would simplify the integrand?
thx

 

[Steven G]

If you had \(\displaystyle \int \dfrac{xdx}{\sqrt{1+x^2}}\), then you can use trig substitution. I would however use u-sub with u =1+x^2

Now if you had \(\displaystyle \int \dfrac {dx}{(x+\sqrt{1+x^2})^2}\), I would use Euler sub.

Try to see if the first integral I just posted would work with Euler-Sub.