What is the integration of cosec2x??

Question

sepeario · Answer

https://www.symbolab.com/solver/integral-calculator/%5Cint%20cosec%5Cleft(2x%5Cright)dx/?origin=enterkey

anonymous · Answer

From Mathematica.
$$\int\limits \csc (2 x) \, dx=\frac{1}{2} \log (\sin (x))-\frac{1}{2} \log (\cos (x)) $$

irishboy123 · Answer

from standard integral: $\int \csc{u} \, du = \ln{\left| \csc{u} - \cot{u}\right|} + C$ https://gyazo.com/75a731bf97950a5f69995fa4255a19f7 so for $ \int \csc{2x} \, dx $ sub $u = 2x, du = 2 dx, dx = du/2$ $\implies \frac{1}{2}\int \csc{u} \, du = \\frac{1}{2} \ln{\left| \csc{u} - \cot{u}\right|} + C \= \frac{1}{2} \ln{\left| \csc{2x} - \cot{2x}\right|} + C$ $csc{2x} - \cot{2x}$ simplifies $\large \frac{1}{sin \ 2x} - \frac{cos \ 2x}{sin \ 2x} = \frac{1 - (1- 2 sin^2x)}{2 \ sinx \ cos x} = tan \ x$ $\implies \int \csc{2x} \, du = \frac{1}{2} \ln{\left| tan \ x\right|} + C$ the sub for the underlying integral follows this original idea for sec https://en.wikipedia.org/wiki/Integral_of_the_secant_function

anonymous · Answer

On the off-chance you meant $\csc^2x$, recall that $\dfrac{d}{dx}\cot x=-\csc^2x$.