试证定积分 ∫(0,1)(x-1)／lnx dx=ln2

elim

题：试证 \(\small\displaystyle\;\int_0^1\frac{x-1}{\ln x}dx = \ln 2.\)

xuke

因为有
\[x-1=x^1-x^0=x^y\mid_{0}^{1}=\ln x\int_0^1{x^y\text{d}y}\]
所以
\begin{align}
\int_0^1{\frac{x-1}{\ln x}\text{d}x}&=\int_0^1{\text{d}x}\int_0^1{x^y\text{d}y}
\\
&=\int_0^1{\text{d}y}\int_0^1{x^y\text{d}x}
\\
&=\int_0^1{\frac{1}{y+1}x^{y+1}\mid_{0}^{1}\text{d}y}
\\
&=\int_0^1{\frac{1}{y+1}\text{d}y}
\\
&=\ln \left| y+1 \right|\mid_{0}^{1}
\\
&=\ln 2
\end{align}

elim

谢谢xuke !  解得漂亮．

ccmmjj

哼哼，真不错。

luyuanhong

楼上 xuke 的解答很好！已收藏。