integral from 0 to ln(16) of tanh(x)

解答

\int_{0}^{l n (16)} tanh (x) d x

ln (\frac{257}{32})

十进制

2.08334 \dots

求解步骤

\int_{0}^{l n (16)} tanh (x) d x

使用三角恒等式改写

= \int_{0}^{l n (16)} \frac{sinh ( x )}{cosh ( x )} d x

使用换元积分法

= \int_{1}^{\frac{257}{32}} \frac{1}{u} d u

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= [ln ∣ u ∣]_{1}^{\frac{257}{32}}

带入上下限计算后得:

ln (\frac{257}{32})

= ln (\frac{257}{32})

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