integrate 1/(x^2+4) without using trigonometric substitution?

Question

no trig substitution, only partial fractions

Answer 1

integral 1/(x^2+4) = integral 1/[(x+2i)(x-2i)]
= (1/4i) integral [ 1/(x-2i)-1/(x+2i) ]
=(1/4i) ( ln(x-2i) -ln(x+2i) )
=(i/4) ln [ (x+2i)/(x-2i) ]

Answer 2

Look at the derivative of arctan(x). There's a substitution that works.

To use partial fractions (and not substitution with arctan), you'll need to factor x^2+4, which can only be done with complex numbers.

Answer 3

1/(x^2+4) = 1/4*(1/((x/2)^2+1)

The int of 1((x/2)^2+1) is 2 Arctan(x/2)
so the result is

1/2 Arctan(x/2)