Deriving integration formulae

Some inquisitive students have asked me before, how the MF15 integration formulas come about. I thought I should share it too then.

So we want to \int\frac {1}{\sqrt{4-x^2}}dx and yes, we know the formula can be plugged directly… but what if we want to avoid the formula. Now actually, the trick here involves your trigonometry identities, along with substitution methods. We have a sin here, so we only know one trigo identity that involves sin and that is sin^{2}x + cos^{2}x = 1. Hmmm, so my approach here will be to let x=2cost. We will see shortly why I chose to put a 2 and that using cost or sint will make no difference. You can try them yourself!

Leibniz: Father of Integration

Credits: Wikipedia

lets first find that dx=2sint dt

\int\frac {1}{\sqrt{4-x^2}}dx= \int \frac {1}{\sqrt{4-4cos^{2}t}}(2sint dt)&s=3

If you notice, this explains why there is a need for us to introduce 2cost instead of just cost.

Having 4-4cos^{2}t allows us to simplify it to 4sin^{2}t

We have \int \frac{1}{\sqrt{4sin^{2}t}}(2sint dt)=\int\frac{1}{2sint}(2sint)dt=\int1dt

Finally, \int1dt=t+C = sin^{-1}(\frac{x}{2}) + C

That was long! But i hope it give you some insights to the formulas.

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