Integration by substitution

[#1procrastinator]

Is there a way to use this formula (imagine the sexy elongated s in front)

f'(g(x))g'(x)dx

rather than the equivalent where you let u=g(x)

f(u)du?

I'm not too comfortable with the differentials

[Mao]

Sure.



Upon first glance, the above may seem straight forward. Give it 5 minutes, and you'll be asking yourself "what the hell is dg?". The infinitesimal is no longer dx, it is now with respect to the function you just substituted. You will need to change your variable (i.e. let g=blah blah). And we're back to the exact same procedure as the other method.

The key point here is, in the substitution method, you can't retain your throughout the whole question. You're going to have to substitute something in place (hence the name).

So the easiest method is actually "let u=blah blah" at the start.

[#1procrastinator]

Take the long way first lol. Thanks a lot Mao