integral of ysinh(y/a)

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende die partielle Integration an

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

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

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

Setze in

u = \frac{y}{a}

ein

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

Vereinfache

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}))

Füge eine Konstante zur Lösung hinzu

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

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