Integrate $\int \csc 2Q\,\mathrm{d}Q$

Question

I need to use

$\cot Q+\tan Q=2\csc 2Q$

to integrate

$$\int \csc 2Q\,\mathrm{d}Q.$$

the integral becomes $$\frac12\int\left(\frac{\cos Q}{\sin Q} + \frac{\sin Q}{\cos Q}\right)\,\mathrm{d}Q$$

Is it possible to use substitution, what are other methods?

Peter · Accepted Answer · 2014-10-21 20:23:55Z

1

Hint : The derivation of sinQ is cosQ and the derivation of cosQ is -sinQ. So you can solve the integral by two substitutions.

answered Oct 21 '14 at 20:23

Peter

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$\begingroup$ How do you turn $1/2[lnsqrt{3} - ln(1/sqrt{3})]$ to 1/2ln3 $\endgroup$ – Arodi007 Oct 21 '14 at 20:34
1

$\begingroup$ @Arodi007 $\ln \sqrt{3} + \ln\frac{1}{\sqrt{3}} = \ln\sqrt{3} + \ln\sqrt{3} = \ln((\sqrt{3}\cdot\sqrt{3}) = \ln 3$. Note that $\ln x+\ln y=\ln(xy)$ and $-\ln x=\ln(1/x)$ hold. (Anyway, Kirino is so cute, isn't it?) $\endgroup$ – Hanul Jeon Oct 21 '14 at 23:32

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