Question

What is the indefinite integral of ∫ [sin (π/x)]/ x^2] dx ?

This is what I did:
Let u = π/x
(to get the derivative and du:)
π*1/x
π(-1/x^2)dx = du
π(1/x^2)dx = (-1)du
so, 1/x^2 = -1/πdu

then ∫ [sin (π/x)]/ x^2] dx = ∫ sin(u) (-1/π)du

= -1/π ∫ sin u du

= -1/π (-cos u) + C

= 1/π (cos u) + C

sub back in:

1/π cos (π/x) + C


I'm unsure of this because I don't know if I got du the right way.

Should it have been

Let u = π/x
-π/x^2 dx = du
1/x^2 dx = -πdu

That would make the answer a lot different...

Help!

Answer 1

Your first derivation is correct.

The last equation you wrote does not follow from (-π/x^2) dx = du

You did it right the first time