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受欢迎的微积分 >

integral of xcsch^2((x^2)/2)

解答

\int x csch^{2} (\frac{x ^{2}}{2}) d x

解答

- coth (\frac{x ^{2}}{2}) + C

求解步骤

\int x csch^{2} (\frac{x ^{2}}{2}) d x

\frac{x ^{2}}{2} = \frac{1}{2} (x^{2} - 2) + 1

= \int x csch^{2} (\frac{1}{2} (x^{2} - 2) + 1) d x

乘开

\frac{1}{2} (x^{2} - 2) + 1 : \frac{1}{2} x^{2}

= \int x csch^{2} (\frac{1}{2} x^{2}) d x

使用换元积分法

= \int \frac{csch ^{2} ( \frac{u}{2} )}{2} d u

提出常数:

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

= \frac{1}{2} \cdot \int csch^{2} (\frac{u}{2}) d u

使用换元积分法

= \frac{1}{2} \cdot \int csch^{2} (v) \cdot 2 d v

提出常数:

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

= \frac{1}{2} \cdot 2 \cdot \int csch^{2} (v) d v

使用积分常见定则:

\int csch^{2} (v) d v = - coth (v)

= \frac{1}{2} \cdot 2 (- coth (v))

代回

= \frac{1}{2} \cdot 2 (- coth (\frac{x ^{2}}{2}))

化简

\frac{1}{2} \cdot 2 (- coth (\frac{x ^{2}}{2})) : - coth (\frac{x ^{2}}{2})

= - coth (\frac{x ^{2}}{2})

解答补常数

= - coth (\frac{x ^{2}}{2}) + C

作图

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