quick question about integration?

Question

I was too sick to go to class today, so I copied my note from a friend of mine, but one of the things in the note was very confusing. here goes,

The question was

integrate cos y / ((sin^2)y + sin y - 6)

sin y = t
cos y dy = dt

I understand up till here.

then, the integration looks like this
dy / t^2 + t -6
= 1/5 integral ( (1/ t-2) - (1/t+3)) dt

then so on.

Where the heck did 1/5 come from?

This section was on partial fraction integration.

Answer 1

First, you made a transcription error. It's dt/(t^2 + t - 6), NOT dy/(t^2 + t - 6).

[Please learn to use parentheses or brackets, to avoid "dangling dividers" which don't indicate where they end. Such things should also be used where there's the SLIGHTEST chance of ambiguity.]

Now t^2 + t - 6 = (t - 2) (t + 3), so it's dt / [ (t - 2) (t + 3) ].

You now want to separate the integrand (containing in the denominator a product of two factors) into the sum or difference of two terms with single factor denominators, in such a way that there's no ' t ' in the numerator. You do this by noting that:

1 / (t - 2) - 1 / (t + 3) = [(t + 3) - (t - 2)] / [ (t - 2) (t + 3) ]

= 5 / [ (t - 2) (t + 3) ].

And, voila, there's your 5! You now have :

dt / [ (t - 2) (t + 3) ] = dt (1/5) [1 / (t - 2) - 1 / (t + 3)].

Your friend's notes were accurate. Get well soon!

QED

Live long and prosper.