What is the integral of ysinh(y/a) ?

The integral of ysinh(y/a) is a^2(1/a ycosh(y/a)-sinh(y/a))+C

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integral of ysinh(y/a)

\int y sinh (\frac{y}{a}) d y

a^{2} (\frac{1}{a} y cosh (\frac{y}{a}) - sinh (\frac{y}{a})) + C

Solution steps

\int y sinh (\frac{y}{a}) d y

Apply u-substitution

: \int a^{2} u sinh (u) d u

= \int a^{2} u sinh (u) d u

Take the constant out:

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

= a^{2} \cdot \int u sinh (u) d u

Apply Integration By Parts

: u cosh (u) - \int cosh (u) d u

= a^{2} (u cosh (u) - \int cosh (u) d u)

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

= a^{2} (u cosh (u) - sinh (u))

Substitute back

u = \frac{y}{a}

= a^{2} (\frac{y}{a} cosh (\frac{y}{a}) - sinh (\frac{y}{a}))

Simplify

a^{2} (\frac{y}{a} cosh (\frac{y}{a}) - sinh (\frac{y}{a})) : a^{2} (\frac{1}{a} y cosh (\frac{y}{a}) - sinh (\frac{y}{a}))

= a^{2} (\frac{1}{a} y cosh (\frac{y}{a}) - sinh (\frac{y}{a}))

Add a constant to the solution

= a^{2} (\frac{1}{a} y cosh (\frac{y}{a}) - sinh (\frac{y}{a})) + C

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What is the integral of ysinh(y/a) ?
The integral of ysinh(y/a) is a^2(1/a ycosh(y/a)-sinh(y/a))+C