What is the integral of sqrt(1+((e^x-e^{-x))/2)^2} ?

The integral of sqrt(1+((e^x-e^{-x))/2)^2} is sinh(x)+C

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integral of sqrt(1+((e^x-e^{-x))/2)^2}

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

sinh (x) + C

Solution steps

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

Rewrite using trig identities

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

Apply Integration By Parts

= x cosh^{2} (x) - \int \frac{x cosh ( x ) sinh ( x )}{cosh ^{2} ( x )} d x

\int \frac{x cosh ( x ) sinh ( x )}{cosh ^{2} ( x )} d x = x cosh (x) - sinh (x)

= x cosh^{2} (x) - (x cosh (x) - sinh (x))

Simplify

x cosh^{2} (x) - (x cosh (x) - sinh (x)) : sinh (x)

= sinh (x)

Add a constant to the solution

= sinh (x) + C

What is the integral of sqrt(1+((e^x-e^{-x))/2)^2} ?
The integral of sqrt(1+((e^x-e^{-x))/2)^2} is sinh(x)+C