# Do I integrate it using Trigonometry substitution?

by caseyjay
Tags: integrate, substitution, trigonometry
 P: 19 Dear all, I would like to evaluate $$\int\frac{1}{1-2sin\left(x\right)}dx$$ Firstly, I make use of the Weierstrass substitution method by letting: $$u=tan\left(\frac{x}{2}\right)$$ and therefore $$sin\left(x\right)=\frac{2u}{1+u^{2}}$$ and $$dx=\frac{2}{1+u^{2}}du$$ Eventually I can rewrite my integral as: $$2\int\frac{1}{u^{2}-4u+1}du$$ Since the denominator of the integrand cannot be factorised, I try using trigonometry substitution by first rewriting the integral as $$2\int\frac{1}{\left(u-2\right)^{2}-\sqrt{3}^{2}}du$$ And then by letting $$u=2+\sqrt{3}sec\left(\theta\right)$$ $$du=\sqrt{3}sec\left(\theta\right)tan\left(\theta\right)d\theta$$ After that I substitute $$u$$ and $$du$$ into the equation and I obtain $$\frac{2}{\sqrt{3}}\int\frac{sec\left(\theta\right)}{tan\left(\theta\rig ht)}d\theta=\frac{2}{\sqrt{3}}\int csc\left(\theta\right)d\theta$$ And that will give me: $$\frac{2}{\sqrt{3}}ln\left|csc(\theta)+cot(\theta)\right|+C$$ But now if I replace $$\theta$$ with $$sec^{-1}\frac{u-2}{\sqrt{3}}$$ I am unable to obtain the answer which is given as: $$\frac{1}{\sqrt{3}}ln\left|\frac{tan\left(\frac{x}{2}\right)-2-\sqrt{3}}{tan\left(\frac{x}{2}\right)-2+\sqrt{3}}\right|+C$$ From the answer given, it seems to me that I should use partial fraction instead of trigonometry substitution. However I am pretty sure if I use trigonometry substitution, I should get the answer. May I know what am I doing wrong here? Thank you very much in advance.
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P: 25,486
Hi caseyjay!
 Quote by caseyjay May I know what am I doing wrong here?
Nothing …

(cosecθ + cotθ)2 = (secθ + 1)2/tan2θ = (secθ + 1)/(secθ - 1)
P: 19
 Quote by tiny-tim Hi caseyjay! Nothing … (cosecθ + cotθ)2 = (secθ + 1)2/tan2θ = (secθ + 1)/(secθ - 1)
Hi Tiny-Tim,

May I know why do you square cosecθ + cotθ? Did I miss out any "squares" in my working? I do not know from where and why did you square cosecθ + cotθ.

P: 253

## Do I integrate it using Trigonometry substitution?

The two outside of the ln can become a power by log rules.
 PF Patron HW Helper Sci Advisor Thanks P: 25,486 yeah … as l'Hôpital says, 2lnx = lnx2

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