INTEGRALE IRRAZIONALE PER SOSTITUZIONE (1) - INTEGRALI

Discover the MATHEMATICS WORKBOOK https://andreailmatematico.it/corsi-m... In this lesson, we'll see how to solve the integral of a particular irrational function by substitution. The case in question is the following: ∫sqrad(x^2 – 25)dx The substitution we adopt is the following: t= x – sqrad(x^2 – 25) Recall that, in general, to solve an integral by substitution, we're dealing with a function of the type: ∫f(g(x))dx First, we proceed with a substitution of the type: f(x) = t At this point, we need to think about the differential. We therefore express x as an (inverse) function of t. x= f^(-1) (t) Then we differentiate both sides of the equation in their respective variables and multiply by the differentials. (x)' dx = (f^(-1) (t))' dt  dx= (f^(-1) (t))' dt So we can rewrite our integral entirely in t. ∫f(g(x))dx = ∫f(t) (f^(-1) (t))' dt Another substitution process that could occur is the following. Let's start with an integral of the form: ∫f(x) dx We proceed with the following substitution, reading x as g(t) x= g(t) Then we work on the differentials: dx= (g(t)'dt At this point, we can rewrite our integral everything with respect to the letter t ∫f(x) dx = ∫f(g(t)) (g(t)’dt) A third type of integral solved by substitution is the following: We start with a composite form of the type ∫f(g(x))dx We opt for the following substitution: g(x) = z(t) If we work immediately on the differentials, g’(x)dx = z’(t)dt we obtain a hybrid form in x and t, so it is necessary to calculate x as a function of t with the inverse function and then work on the differential. g(x) = z(t)  x=g^(-1) (z(t))  dx= (g^(-1) (z(t)))’dt At this point, the final integral becomes: ∫f(g(x))dx= ∫f(z(t)) · (g^(-1) (z(t)))’dt Access link to all MATHEMATICS COURSES https://andreailmatematico.it/corsi-m... -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- To learn more, you can visit my website here https://andreailmatematico.it/ Subscribe to my channel here    / andreailmatematico   Visit my website https://andreailmatematico.it/ Follow me on Facebook   / lamatematicadiandreailmatematico