What is the integral from 0 to a of arcsinh(t) ?

The integral from 0 to a of arcsinh(t) is aarcsinh(a)-sqrt(a^2+1)+1

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integral from 0 to a of arcsinh(t)

\int_{0}^{a} arcsinh (t) d t

a arcsinh (a) - a^{2} + 1 + 1

Solution steps

\int_{0}^{a} arcsinh (t) d t

Apply Integration By Parts

= [t arcsinh (t) - \int \frac{t}{t ^{2} + 1} d t]_{0}^{a}

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

= [t arcsinh (t) - t^{2} + 1]_{0}^{a}

Compute the boundaries:

a arcsinh (a) - a^{2} + 1 + 1

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

What is the integral from 0 to a of arcsinh(t) ?
The integral from 0 to a of arcsinh(t) is aarcsinh(a)-sqrt(a^2+1)+1