- #1
donaldparida
- 146
- 10
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.
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.