# 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:

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