integral of ysinh(y/a)

解答

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

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

求解步骤

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

使用换元积分法

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

提出常数:

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

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

使用分布积分法

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

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

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

u = \frac{y}{a}

代回

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

化简

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}))

解答补常数

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

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