What is the integral of sinh^2(t) ?

The integral of sinh^2(t) is 1/2 (-t+1/2 sinh(2t))+C

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integral of sinh^2(t)

\int sinh^{2} (t) d t

\frac{1}{2} (- t + \frac{1}{2} sinh (2 t)) + C

Solution steps

\int sinh^{2} (t) d t

Rewrite using trig identities

= \int \frac{- 1 + cosh ( 2 t )}{2} d t

Take the constant out:

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

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

Apply the Sum Rule:

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

= \frac{1}{2} (- \int 1 d t + \int cosh (2 t) d t)

\int 1 d t = t

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

= \frac{1}{2} (- t + \frac{1}{2} sinh (2 t))

Add a constant to the solution

= \frac{1}{2} (- t + \frac{1}{2} sinh (2 t)) + C

What is the integral of sinh^2(t) ?
The integral of sinh^2(t) is 1/2 (-t+1/2 sinh(2t))+C