一道有趣的定积分
设$\varphi = {\sqrt{5}+1}/{2}$,求证:\[\int_0^\infty\frac{1}{(1+x^\varphi)^\varphi}\mathrm dx = 1\] 可以试着做变量替换\(y=1+x^{\varphi}\),于是积分变成
\[\int_1^{\infty}\frac{dy}{\varphi (y-1)^{1-\frac{1}{\varphi}}y^{\varphi}}\]
继续做替换\(z=1/y\),我们可以得到
\[\int_0^{1}\frac{dz}{\varphi (1-z)^{1-\frac{1}{\varphi}}}\]
于是最后变成\[\left.-(1-z)^{\frac{1}{\varphi}}\middle|_0^1\right.=1\]
页:
[1]