How to restate the gamma function?

Question

If we take this definition for the gamma function:

$$ \Gamma(\alpha)=\int_{0}^{\infty}x^{\alpha-1}e^{-x}\mathrm dx \ \text{ for } \ \alpha>0,$$

and we substitute $\lambda y$ for $x$. Why do we get: $$\Gamma(\alpha)=\lambda^{\alpha}\int_{0}^{\infty}y^{\alpha-1}e^{-\lambda y}\mathrm dy$$ and not: $$\Gamma(\alpha)=\lambda^{\alpha - 1}\int_{0}^{\infty}y^{\alpha-1}e^{-\lambda y}\mathrm dy$$

See (https://www.probabilitycourse.com/chapter4/4_2_4_Gamma_distribution.php)

Answer 1

When substituting, $\lambda y = x$ means $\lambda\mathrm dy = \mathrm dx$. That is where the extra $\lambda$ comes from.