integral of 1/(ln^3(x))

解答

\int \frac{1}{ln ^{3} ( x )} d x

- \frac{x}{2 ln ^{2} ( x )} + \frac{1}{2} (- \frac{x}{ln ( x )} + Ei (ln (x))) + C

求解步骤

\int \frac{1}{ln ^{3} ( x )} d x

使用换元积分法

= \int \frac{e ^{u}}{u ^{3}} d u

使用分布积分法

= - \frac{e ^{u}}{2 u ^{2}} - \int - \frac{e ^{u}}{2 u ^{2}} d u

\int - \frac{e ^{u}}{2 u ^{2}} d u = - \frac{1}{2} (- \frac{e ^{u}}{u} + Ei (u))

= - \frac{e ^{u}}{2 u ^{2}} - (- \frac{1}{2} (- \frac{e ^{u}}{u} + Ei (u)))

u = ln (x)

代回

= - \frac{e ^{l n (x)}}{2 ln ^{2} ( x )} - (- \frac{1}{2} (- \frac{e ^{l n (x)}}{ln ( x )} + Ei (ln (x))))

化简

- \frac{e ^{l n (x)}}{2 ln ^{2} ( x )} - (- \frac{1}{2} (- \frac{e ^{l n (x)}}{ln ( x )} + Ei (ln (x)))) : - \frac{x}{2 ln ^{2} ( x )} + \frac{1}{2} (- \frac{x}{ln ( x )} + Ei (ln (x)))

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

解答补常数

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

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