integral of sech(ln(x))

解答

\int sech (ln (x)) d x

x sech (ln (x)) - sech (ln (x)) arcsech (sech (ln (x))) - 2 arcsin (\frac{1}{2} sech (ln (x)) + 1) + C

求解步骤

\int sech (ln (x)) d x

使用分布积分法

= x sech (ln (x)) - \int - sech (ln (x)) tanh (ln (x)) d x

\int - sech (ln (x)) tanh (ln (x)) d x = sech (ln (x)) arcsech (sech (ln (x))) + 2 arcsin (\frac{1}{2} sech (ln (x)) + 1)

= x sech (ln (x)) - (sech (ln (x)) arcsech (sech (ln (x))) + 2 arcsin (\frac{1}{2} sech (ln (x)) + 1))

化简

= x sech (ln (x)) - sech (ln (x)) arcsech (sech (ln (x))) - 2 arcsin (\frac{1}{2} sech (ln (x)) + 1)

解答补常数

= x sech (ln (x)) - sech (ln (x)) arcsech (sech (ln (x))) - 2 arcsin (\frac{1}{2} sech (ln (x)) + 1) + C

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