What is the integral of arccsc(2x) ?

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

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integral of arccsc(2x)

\int arccsc (2 x) d x

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

Solution steps

\int arccsc (2 x) d x

Apply u-substitution

: \int arccsc (u) \frac{1}{2} d u

= \int arccsc (u) \frac{1}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int arccsc (u) d u

Apply Integration By Parts

: u arccsc (u) - \int - \frac{u}{u ^{2} u ^{2} - 1} d u

= \frac{1}{2} (u arccsc (u) - \int - \frac{u}{u ^{2} u ^{2} - 1} d u)

\int - \frac{u}{u ^{2} u ^{2} - 1} d u = - ln u + u^{2} - 1

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

Substitute back

u = 2 x

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

Simplify

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

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

Add a constant to the solution

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

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