Question

$$\int_0^\infty\int_0^\infty\int_0^\infty\frac{(xyz)^{-1/7}(yz)^{-1/7}z^{-1/7}}{(x+1)(y+1)(z+1)}\:dx\,dy\,dz$$

İntegralini çözün.

Answer 1

İntegralimiz :

$$\int_0^\infty\int_0^\infty\int_0^\infty\frac{(xyz)^{-1/7}(yz)^{-1/7}z^{-1/7}}{(x+1)(y+1)(z+1)}\:dx\,dy\,dz$$

İntegrali $3$ parçaya ayıralım.

$$\Bigg(\int_0^\infty\:\frac{x^{-1/7}}{x+1}\:dx\Bigg)\:\Bigg(\int_0^\infty\:\frac{y^{-2/7}}{y+1}\:dy\Bigg)\:\Bigg(\int_0^\infty\:\frac{z^{-3/7}}{z+1}\:dz\Bigg)$$

Buradaki eşitliği kullanarak integralleri bulalım.

$$\pi^3\:\csc\bigg(\frac{6\pi}{7}\bigg)\csc\bigg(\frac{5\pi}{7}\bigg)\csc\bigg(\frac{4\pi}{7}\bigg)$$

Gerekli sadeleştirmeleri yaparak cevaba ulaşabiliriz.

$$\color{#A00000}{\boxed{\int_0^\infty\int_0^\infty\int_0^\infty\frac{(xyz)^{-1/7}(yz)^{-1/7}z^{-1/7}}{(x+1)(y+1)(z+1)}\:dx\,dy\,dz=\frac{8\sqrt{7}}{7}\,\pi^3\approx93,75416820}}$$