What is the integral of arccsc(x/2) ?

The integral of arccsc(x/2) is xarccsc(x/2)+2ln(1/2 |sqrt(x^2-4)+x|)+C

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integral of arccsc(x/2)

\int arccsc (\frac{x}{2}) d x

x arccsc (\frac{x}{2}) + 2 ln (\frac{1}{2} x^{2} - 4 + x) + C

Solution steps

\int arccsc (\frac{x}{2}) d x

Apply Integration By Parts

= x arccsc (\frac{x}{2}) - \int - \frac{2 x}{x ^{2} x ^{2} - 4} d x

\int - \frac{2 x}{x ^{2} x ^{2} - 4} d x = - 2 ln (\frac{1}{2} x^{2} - 4 + x)

= x arccsc (\frac{x}{2}) - (- 2 ln (\frac{1}{2} x^{2} - 4 + x))

Simplify

= x arccsc (\frac{x}{2}) + 2 ln (\frac{1}{2} x^{2} - 4 + x)

Add a constant to the solution

= x arccsc (\frac{x}{2}) + 2 ln (\frac{1}{2} x^{2} - 4 + x) + C

What is the integral of arccsc(x/2) ?
The integral of arccsc(x/2) is xarccsc(x/2)+2ln(1/2 |sqrt(x^2-4)+x|)+C