What is the integral of sinh(ax) ?

The integral of sinh(ax) is 1/a cosh(ax)+C

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integral of sinh(ax)

\int sinh (a x) d x

\frac{1}{a} cosh (a x) + C

Solution steps

\int sinh (a x) d x

Apply u-substitution

= \int \frac{sinh ( u )}{a} d u

Take the constant out:

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

= \frac{1}{a} \cdot \int sinh (u) d u

Use the common integral:

\int sinh (u) d u = cosh (u)

= \frac{1}{a} cosh (u)

Substitute back

u = a x

= \frac{1}{a} cosh (a x)

Add a constant to the solution

= \frac{1}{a} cosh (a x) + C