Question

integral of dx/(2 + 2 cos(x) - 2sqrt(3)sin(x))

Answer 1

You have 2 terms in this integrand, separated by a negative sign. The two terms have no relationship to each other that I can see. So your best bet may be to attempt to integrate each of the 2 terms separately.

What is Int(dx/(2 + 2 cos(x) ?

or\[\int\limits_{}^{}\frac{ dx }{ 2(1+\cos x) }?\] One approach that might work and is worth investigating: Mult. numerator and denom. of this integrand by the conjugage of (1+cos x); that is, mult. by (1-cos x).

Answer 2

HI!!

Answer 3

does it help to know that \[\cos(x)+\sqrt{3}\sin(x)= 2\sin(\frac{\pi}{6}-x)\] or even \[-2\sin(x-\frac{\pi}{6})\]

Answer 4

oops i think i am off by a signe

Answer 5

no, just made a typo \[cos(x)-\sqrt3\sin(x)=-2\sin(x-\frac{\pi}{6})\]

Answer 6

Building on the above one can write that
\[
\frac{1}{-2 \sqrt{3} \sin (x)+2 \cos (x)+2}=\frac{1}{2 \left(1-2 \sin
\left(x-\frac{\pi }{6}\right)\right)}
\]

Answer 7

to check you answer,
\[
\int \frac{1}{2 \left(1-2 \sin \left(x-\frac{\pi }{6}\right)\right)} \,
dx=\frac{\tanh ^{-1}\left(\frac{\tan \left(\frac{1}{12} (\pi -6
x)\right)+2}{\sqrt{3}}\right)}{\sqrt{3}}
\]

Answer 8

The above gives a step by step solution