integral from 0 to 0.5 of 1/(xln^3|x|)

解答

\int_{0}^{0.5} \frac{1}{x ln ^{3} ∣ x ∣} d x

- 1.04068 \dots

求解步骤

\int_{0}^{0.5} \frac{1}{x ln ∣ x ∣ ^{3}} d x

拿掉绝对值

= \int_{0}^{0.5} \frac{1}{x ln ^{3} ( x )} d x

使用分布积分法

= \frac{1 - ln ^{3} ( x ) \cdot \int - \frac{l n ( x ) + 3}{x l n ^{4} ( x )} d x}{ln ^{3} ( x )}_{0}^{0.5}

\int - \frac{ln ( x ) + 3}{x ln ^{4} ( x )} d x = \frac{1}{2 ln ^{2} ( x )} + \frac{1}{ln ^{3} ( x )}

= \frac{1 - ln ^{3} ( x ) ( \frac{1}{2 l n ^{2} ( x )} + \frac{1}{l n ^{3} ( x )} )}{ln ^{3} ( x )}_{0}^{0.5}

\frac{1 - ln ^{3} ( x ) ( \frac{1}{2 l n ^{2} ( x )} + \frac{1}{l n ^{3} ( x )} )}{ln ^{3} ( x )} = - \frac{0.5}{ln ^{2} ( x )}

= [- \frac{0.5}{ln ^{2} ( x )}]_{0}^{0.5}

带入上下限计算后得:

- 1.04068 \dots

= - 1.04068 \dots

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