What is the integral of (e^s-e^{-s})^2 ?

The integral of (e^s-e^{-s})^2 is -2s+sinh(2s)+C

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integral of (e^s-e^{-s})^2

\int (e^{s} - e^{- s})^{2} d s

- 2 s + sinh (2 s) + C

Solution steps

\int (e^{s} - e^{- s})^{2} d s

Rewrite using trig identities

= \int (2 sinh (s))^{2} d s

Simplify

(2 sinh (s))^{2} : 4 sinh^{2} (s)

= \int 4 sinh^{2} (s) d s

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int sinh^{2} (s) d s

Rewrite using trig identities

= 4 \cdot \int \frac{- 1 + cosh ( 2 s )}{2} d s

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \frac{1}{2} \cdot \int - 1 + cosh (2 s) d s

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 4 \cdot \frac{1}{2} (- \int 1 d s + \int cosh (2 s) d s)

\int 1 d s = s

\int cosh (2 s) d s = \frac{1}{2} sinh (2 s)

= 4 \cdot \frac{1}{2} (- s + \frac{1}{2} sinh (2 s))

Simplify

4 \cdot \frac{1}{2} (- s + \frac{1}{2} sinh (2 s)) : - 2 s + sinh (2 s)

= - 2 s + sinh (2 s)

Add a constant to the solution

= - 2 s + sinh (2 s) + C

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What is the integral of (e^s-e^{-s})^2 ?
The integral of (e^s-e^{-s})^2 is -2s+sinh(2s)+C