Question

whats wrong with this substitution

Answer 1

\[\int_0^{\infty} \frac{\sin(ax)}{x}~dx\]

sub \( u = ax \implies dx = \dfrac{du}{a}\) and \(\dfrac{1}{x} = \dfrac{a}{u} \)

\[\int_0^{\infty} \frac{\sin(u)}{u}~du\]

Answer 2

nothing

Answer 3

maybe the bounds redefined depending on possibilities of a

Answer 4

what happened to \(a\)

Answer 5

got cancelled with du/a

Answer 6

does that mean the value of integral doesn't depend on \(a\) ?

if so how can we explain this ?

\[\int_0^{\infty} \frac{\sin(2x)}{x}~dx \ne\int_0^{\infty} \frac{\sin(-2x)}{x}~dx \]

Answer 7

it does depend on a

Answer 8

depending on a the integral boudns can change

Answer 9

sin(-t) = -sin(t)
so clearly the two integrals differ by a sign

Answer 10

as long as a>0 its not a problem though

Answer 11

why the restriction a>0 ?

Answer 12

it was just straightforward regular substitution right ?

Answer 13

because