1. ## Weird Integral

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)

2. Originally Posted by rainbowkaine

2. Find dy/dx of y=ln(x^2+y^2)
Use implicit differentiation.

$\displaystyle y' = \left( \frac{1}{x^2+y^2} \right) \left(2x+2y*y' \right)$

From here, expand the right hand side and then collect all terms containing y' onto one side of the equation, factor out y' and solve algebraically.

3. Originally Posted by rainbowkaine
1. Find the indefinite integral of dx/(x(x^2+4)^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