What is the integral of arccosh(2x) ?

The integral of arccosh(2x) is xarccosh(2x)-1/2 sqrt((2x-1)(2x+1))+C

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integral of arccosh(2x)

\int arccosh (2 x) d x

x arccosh (2 x) - \frac{1}{2} (2 x - 1) (2 x + 1) + C

Solution steps

\int arccosh (2 x) d x

Apply u-substitution

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

= \int arccosh (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 arccosh (u) d u

Apply Integration By Parts

: u arccosh (u) - \int \frac{u}{u - 1 u + 1} d u

= \frac{1}{2} (u arccosh (u) - \int \frac{u}{u - 1 u + 1} d u)

\int \frac{u}{u - 1 u + 1} d u = (u - 1) (u + 1)

= \frac{1}{2} (u arccosh (u) - (u - 1) (u + 1))

Substitute back

u = 2 x

= \frac{1}{2} (2 x arccosh (2 x) - (2 x - 1) (2 x + 1))

Simplify

\frac{1}{2} (2 x arccosh (2 x) - (2 x - 1) (2 x + 1)) : x arccosh (2 x) - \frac{1}{2} (2 x - 1) (2 x + 1)

= x arccosh (2 x) - \frac{1}{2} (2 x - 1) (2 x + 1)

Add a constant to the solution

= x arccosh (2 x) - \frac{1}{2} (2 x - 1) (2 x + 1) + C

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What is the integral of arccosh(2x) ?
The integral of arccosh(2x) is xarccosh(2x)-1/2 sqrt((2x-1)(2x+1))+C