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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: #1procrastinator on July 27, 2012, 12:37:40 am

Title: Integration by substitution
Post by: #1procrastinator on July 27, 2012, 12:37:40 am: 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
Title: Re: Integration by substitution
Post by: Mao on July 27, 2012, 03:45:37 am: Sure.

$\int_{x_0}^{x_1} k\, f'(g(x))g'(x)\, dx = k \int_{g(x_0)}^{g(x_1)} f'(g)\, dg$

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 $x$ 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.
Title: Re: Integration by substitution
Post by: #1procrastinator on July 27, 2012, 02:40:46 pm: Take the long way first lol. Thanks a lot Mao