integral from 1 to e of (ln(x))/(x^3)

解答

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

- \frac{3}{4 e ^{2}} + \frac{1}{4}

十进制

0.14849 \dots

求解步骤

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

使用分布积分法

= [\frac{- ln ( x ) - 2 x ^{2} \cdot \int - \frac{1}{2 x ^{3}} d x}{2 x ^{2}}]_{1}^{e}

\int - \frac{1}{2 x ^{3}} d x = \frac{1}{4 x ^{2}}

= [\frac{- ln ( x ) - 2 x ^{2} \frac{1}{4 x ^{2}}}{2 x ^{2}}]_{1}^{e}

\frac{- ln ( x ) - 2 x ^{2} \frac{1}{4 x ^{2}}}{2 x ^{2}} = \frac{- 2 ln ( x ) - 1}{4 x ^{2}}

= [\frac{- 2 ln ( x ) - 1}{4 x ^{2}}]_{1}^{e}

带入上下限计算后得:

- \frac{3}{4 e ^{2}} + \frac{1}{4}

= - \frac{3}{4 e ^{2}} + \frac{1}{4}

d er i v a t i v e f (θ) = 3 tan (θ)

x \to + (- 3) lim (5)

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