# Integration by substitution

## Summary and examples

In the topic Integration of standard funtions the primitive functions of a restricted number of functions are given.

The primitive functions of most functions cannot easily be calculated.
However, there are also special methods to calculate a primitive function. These methods can be quite complex, but here we focus on a special category, namely substitution.

The functions that we want to discuss now can be written as: met If is the primitive function of then: is the primitive function of: We can verify whether this is correct by differentiating the primitive function, using the chain rule: And indeed, this result is correct.
The word substitution is used because the result above can also be calculated by the following substitution\: and then differentiating : The following examples show how the method of substitution works.

##### Example 1

Solve: The function: has the primitive function: We have for the integrand of the integral: and thus the primitive function of this function is: and thus: The second method gives the same result.

Take: Differentiating this function yields: or: If we subtitute these results in the original integral, we get: If we ‘transform back’: It is a matter of personal preference which of the methods is chosen. In the following examples we will choose either method.

##### Example 2

Solve: The primitive of: is equal to: and thus: ##### Example 3

Solve: We write: and the primitive of: is equal to: So we can write: ##### Example 4

Solve: We write this integral as: The primitive of: is: and thus: ##### Example 5

Solve: The primitive of: is equal to: and thus: 0