I am tutoring a student in Calculus 2 and cannot seem to get the following 2 problems (it's been a while since I took this course)
1. Find the indefinite integral of dx/(x(x^2+4)^2)
2. Find dy/dx of y=ln(x^2+y^2)
hard way: do a trig substitution of $\displaystyle x = 2 \tan \theta$
an easier way: note that we have this integral (by multiplying by x/x): $\displaystyle \int \frac {x}{x^2(x^2 + 4){\color{red}^2}}~dx$
now, let $\displaystyle u = x^2 + 4 \implies \boxed{x^2 = u - 4}$
then $\displaystyle du = 2x~dx$
$\displaystyle \Rightarrow \frac 12 ~du = x~dx$
So our integral becomes:
$\displaystyle \frac 12 \int \frac 1{u^2(u - 4)}~du$
which is relatively easy to do via partial fractions