Integration by Substitution

The method of integration by substitution or change of variable is based on the derivative of the composite function or chain rule.

Integration by Substitution

To change the variable, identify the part of the function that is going to integrate with a new variable, t, in order to obtain a simpler integral.

Steps to Integrate by Substitution

Integral

1.Perform the Change of variable and differentiate the two terms:

Change of Variable

Differentiate

Work out the value of u and dx by substituting these values into the integral.

Substitute the Integral

2. If the resulting integral is simpler, integrate:

Integral

3. Return to the initial variable:

Initial Variable


Example

Integral Example

Integral Operations

Integral Operations

Integral Operations

Integral Operations

Integral Operations

Integral Solution


Usual Change of Variable

1. Change of Variable

2. Change of Variable

3. Change of Variable

4. Change of Variable

5. In the rational functions of radicals with different indices and the same linear radicand, ax + b, the change of variable is t raised to the least common multiple of the indices.

6. If Rational is even:

Change of Variable

7. If racional que una métrica paris not even:

Change of Variable


Examples

Integral Example

Change of Variable

Change of Variable

Integral Operations

Integral Operations

Integral Operations

Integral Operations

Integral Operations

Integral Operations

Integral Operations

Integral Solution

Integral Example

Change of Variable

Change of Variable

Integral Solution

Integral Example

Change of Variable

Change of Variable

Integral Solution

Integral Example

Change of Variable

Change of Variable

Integral Operations

Integral Operations

Integral Solution

Integral Example

Change of Variable

Change of Variable

Integral Operations

Integral Operations

Integral Operations

Integral Solution

Integral Example

Change of Variable

Change of Variable

Integral Operations

Integral Operations

Integral Operations

Integral Operations

cambie variable

Integral Solution