Question

evaluate making u-sub, integrate dx/9+4x^2

Answer 1

i dont think this can be done with u substitution. you have to use trig substitution i believe.

Answer 2

\[\large \int\limits \frac{dx}{4x^2+9}\]Factoring a 9 out of each term from the bottom gives us,\[\large \frac{1}{9}\int\limits\limits \frac{dx}{\left(\dfrac{4}{9}x^2+1\right)}\]

Let \(\large x=\dfrac{3}{2}u\). Taking the derivative of our substitution gives us, \(\large dx=\dfrac{3}{2}du\).

Plugging in our substitution,\[\large \frac{1}{9}\int\limits\limits\limits \frac{\frac{3}{2}du}{\left(u^2+1\right)}\]Factor out the 3/2's and you might recognize it as one of the derivatives you've probably memorized at this point.\[\large \frac{1}{6}\int\limits \frac{1}{u^2+1}du\] As Brandon pointed out, it's better to just apply a Trigonometric Substitution from the start. But if you haven't learned that yet, then your goal is to get it into this form, and match it up with one of the integrals from your notes.