Test data: Latex - Integrals in comparison

By syntax

\begin{gather}
\Gamma(z)=\int_0^\infty \!\! e^{-x}x^{z-1} \mathrm{d}x
\quad\text{vs.}
\quad \Gamma(z)=\int\limits_0^\infty e^{-x}x^{z-1}\mathrm{d}x \\
B(x,y) = \int_0^\infty \!\! \frac{t^{x-1}}{(1+t)^{x+y}}\mathrm{d}t
\quad\text{vs.}\quad
B(x,y) = \int\limits _0^\infty \frac{t^{x-1}}{(1+t)^{x+y}}\mathrm{d}t \\
\int_{-\infty}^\infty \frac{\frac{a(x)}{b(x)}}{\frac{c(x)}{f(x)}}\mathrm{d}x 
\quad\text{vs.}\quad
\int\limits_{-\infty}^\infty \frac{\frac{a(x)}{b(x)}}{\frac{c(x)}{f(x)}}\mathrm{d}x
\end{gather}

By render

Γ(z)=∫0∞ ⁣ ⁣e−xxz−1dxvs.Γ(z)=∫0∞e−xxz−1dxB(x,y)=∫0∞ ⁣ ⁣tx−1(1+t)x+ydtvs.B(x,y)=∫0∞tx−1(1+t)x+ydt∫−∞∞a(x)b(x)c(x)f(x)dxvs.∫−∞∞a(x)b(x)c(x)f(x)dx\begin{gather}
\Gamma(z)=\int_0^\infty \!\! e^{-x}x^{z-1} \mathrm{d}x
\quad\text{vs.}
\quad \Gamma(z)=\int\limits_0^\infty e^{-x}x^{z-1}\mathrm{d}x \\
B(x,y) = \int_0^\infty \!\! \frac{t^{x-1}}{(1+t)^{x+y}}\mathrm{d}t
\quad\text{vs.}\quad
B(x,y) = \int\limits _0^\infty \frac{t^{x-1}}{(1+t)^{x+y}}\mathrm{d}t \\
\int_{-\infty}^\infty \frac{\frac{a(x)}{b(x)}}{\frac{c(x)}{f(x)}}\mathrm{d}x 
\quad\text{vs.}\quad
\int\limits_{-\infty}^\infty \frac{\frac{a(x)}{b(x)}}{\frac{c(x)}{f(x)}}\mathrm{d}x
\end{gather}

Resource

tex.stackexchange - \int with \limits or without?