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anonymous
 one year ago
What is the integration of cosec2x??
anonymous
 one year ago
What is the integration of cosec2x??

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0From Mathematica. \[\int\limits \csc (2 x) \, dx=\frac{1}{2} \log (\sin (x))\frac{1}{2} \log (\cos (x)) \]

IrishBoy123
 one year ago
Best ResponseYou've already chosen the best response.1from 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
 one year ago
Best ResponseYou've already chosen the best response.0On the offchance you meant \(\csc^2x\), recall that \(\dfrac{d}{dx}\cot x=\csc^2x\).
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