Methods of integration: direct and indirect substitution

[donaldparida]

I have seen two approaches to the method of integration by substitution (in two different books). On searching the internet i came to know that Approach I is known as the method of integration by direct substitution whereas Approach II is known as the method of integration by indirect substitution.

Approach I

Let I=∫f(φ(x))φ'(x)dx

Let z=φ(x)

∴φ'(x)dx=dz

∴I=∫f(z)dz

Approach II

Let I=∫f(x)dx

Let x=φ(z)

∴dx=φ'(z)dz

∴I=∫f(φ(z))φ'(z)dz

My problem: While i can understand Approach I, I cannot understand Approach II. What is the difference between the two approaches. What is the difference in their usage. I very confused. Please help.

 

[mathman]

Approach II seems to be approach I in reverse order. It is possible that it (II) is meaningful for a specific problem, but otherwise it doesn't seem to have much point. Approach I is widely used.

 

[Stephen Tashi]

Approach I is widely used.

Approach II could be interpreted as what a calculus student encounters. The student must find a useful $\phi(z)$.

whereas Approach II is known as the method of integration by indirect substitution.

To me (in the USA) "integration by indirect substitution" is not commonly used terminology. I find this web page https://www.askiitians.com/iit-jee-indefinite-integral/indirect-substitution/ where the terminology is used. (It also uses "integral" to mean "antiderivative".) The method presented on that page could be in interpreted as your Approach I instead of Approach II.

 

[donaldparida]

@Stephen Tashi , I read about it here: http://math.feld.cvut.cz/mt/txtd/3/txe3da3b.htm.

 

[Stephen Tashi]

If we are finding $\int H(x)dx$ and wish to make a substitution to introduce a new variable $z$, we can write the relation between $x$ and $z$ in two equivalent ways. We can write it as $g(z) = x$ or $g^{-1}(x) = z$ where $g^{-1}$ is the inverse function of $g$.

The two Approaches differ in how they choose to express the same relationship. Visualize the $\phi$ in Approach I as denoting the inverse function of the $\phi$ in Approach II and you can see the approaches are doing the same thing mathematically. They both replace $x$ by some function of another variable.

The two Approaches differ as techniques of pattern recognition. When $H(x)$ is an expression we recognize as containing the product of a function with its derivative such as $(... (sin(x))^2 + ...) ( cos(x))$ we try the substitution $z = sin(x)$. When $H(x)$ is not obviously of that form we might try the substitution $x = arcsin(z)$ just to see what happens. The equations $z = sin(x)$ and $x = arcsin(z)$ express the same substitution.