What is the integral of arcsinh(2x) ?

The integral of arcsinh(2x) is xarcsinh(2x)-1/2 sqrt(4x^2+1)+C

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integral of arcsinh(2x)

\int arcsinh (2 x) d x

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

Solution steps

\int arcsinh (2 x) d x

Apply Integration By Parts

: x arcsinh (2 x) - \int \frac{2 x}{4 x ^{2} + 1} d x

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

\int \frac{2 x}{4 x ^{2} + 1} d x = \frac{1}{2} 4 x^{2} + 1

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

Add a constant to the solution

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

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What is the integral of arcsinh(2x) ?
The integral of arcsinh(2x) is xarcsinh(2x)-1/2 sqrt(4x^2+1)+C