What is the integral of arcsinh(x) ?

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

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integral of arcsinh(x)

\int arcsinh (x) d x

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

Solution steps

\int arcsinh (x) d x

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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