B Methods of integration: direct and indirect substitution

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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

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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

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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.
 

Stephen Tashi

Science Advisor
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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.
 

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