# Motivation behind substitutions in an integral 1

I was reading a textbook on Integration where I came across suggested substitutions for certain types of Integrals. These were as follows: 

Integrals of the form $$\int\dfrac{dx}{(ax+b)\sqrt{px^2+qx+r}}$$ Put $ax+b=\dfrac 1t$



Integrals of the form $$\int\dfrac{dx}{(ax^2+b)\sqrt{cx^2+d}}$$ Put $x=\dfrac 1t$

I cannot understand the intuition/motivation behind these substitutions. I know that these work (I had tried them with a few examples). However these substitutions would never strike me logically. Typically we use substitutions which would simplify the Integral. However, until actually transforming the Integrand, I cannot understand why these substitutions are useful - thus they do not feel intuitive to me and would never strike me ad hoc.Could somebody please explain to me the logic behind using these substitutions? As in without first modifying the integrand and seeing whether the modified integral is easier, how could it be understood that these substitutions would actually work and simplify the Integrand?PS. Perhaps the integrals above are meant to be converted into the following integrals (whose closed forms are known to me):

$\int \dfrac{dx}{x^2+a^2}, \int \dfrac{dx}{x^2-a^2},\int \dfrac{dx}{a^2-x^2},\int \dfrac{dx}{\sqrt{x^2+a^2}},\int \dfrac{dx}{\sqrt{a^2-x^2}},\int \dfrac{dx}{\sqrt{x^2-a^2}},\int\sqrt{a^2-x^2}dx$$\int\sqrt{a^2-x^2}dx,\int\sqrt{a^2+x^2}dx,\int\sqrt{x^2-a^2}dx$