# Integration using trig subs having trouble switching back to the original variable

1. Oct 4, 2004

### Odyssey

Hi, I carried out the integration until the very end...I don't know how to convert the variable back to the original one.

$$\int_{R_{0}}^{R(\Theta)}\frac{du}{u\sqrt{a^2-u^2}}$$

$$Let u = a\sin{\Theta}$$
$$du = a\cos{\Theta}d\Theta$$

The integral becomes...

$$\int_{R_{0}}^{R(\Theta)}\frac{a\cos{\Theta}d\Theta}{a\cos{\Theta}a\sin{\Theta}}$$
$$\frac{1}{a}\int_{R_{0}}^{R(\Theta)}\csc{\Theta}d{\Theta}$$

$$\csc{\Theta}d{\Theta} = \ln {|\csc{\Theta}-\cot{\Theta}|}+C$$

This is where I'm stuck. I don't know how to convert the thetas back into the "u"s. I haven't multiplied the answer by 1/a yet. I know that $$\Theta=\sin^{-1}{u/a}$$, but if I plug the $$\sin^{-1}{u/a}$$ into Theta, the expression becomes super messy and I really don't know what to do with it.

Please help, thanks in advance!

Last edited: Oct 4, 2004
2. Oct 4, 2004

### vsage

make a triangle.. it's the only way. For example, if theta = arcsin(u/a) then sin(arcsin(u/a)) = u/a, cos(arcsin(u/a)) = sqrt(a^2-u^2)/a. Make a triangle with sides u, sqrt(a^2-u^2) and a. You can find all the trig functions from it (be sure to label theta)

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