Documentation

# `intlib`::`changevar`

Change of variable

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## Syntax

```intlib::changevar(`integral`, `eq`, <`var`>)
```

## Description

`intlib::changevar(integral, eq)` performs a change of variable for indefinite and definite integrals.

Mathematically, the substitution rule is formally defined for indefinite integrals as and for definite integrals as `intlib::changevar(integral, eq)` performs in `integral` the change of variable defined by `eq` and returns an unevaluated new integral. You can use the `eval` command to find the closed form of this new integral providing that the closed form exists.

`intlib::changevar` works for indefinite as well as for definite integrals.

The first argument should contain a symbolic integral of type `"int"`. Such an expression can be obtained with `hold` or `freeze`. See Example 1.

If more than two variables occur in `eq`, the new variable must be given as third argument.

If MuPAD® cannot solve the given equation `eq` an error will occur.

## Examples

### Example 1

As a first example we perform a change of variable for the integral . By using the `hold` function we ensure that the first argument is of type `"int"`:

```intlib::changevar(hold(int)(f(x + c), x = a..b), t = x + c, t)```
` `

Note that in this case the substitution equation has two further variables besides x. Thus it is necessary to specify the new integration variable as third argument.

### Example 2

In the following example we use the change of variable method for solving the integral . First we perform the transformation t = ln(x):

```f1 := intlib::changevar(hold(int)(cos(ln(x)), x), t = ln(x), t)```
` `

Now we can evaluate the integral with the MuPAD integrator:

`f2:=eval(f1)`
` `

Finally we change the variable t back to x and get the result:

`F := simplify(f2 | t = ln(x))`
` `

We can also verify the solution of the integral:

`simplify(diff(F,x) - cos(ln(x)))`
` `

## Parameters

 `integral` The integral: an arithmetical expression containing a symbolic `"int"` call `eq` Equation defining the new integration variable in terms of the old one: an equation `var` The new integration variable: an identifier

## Return Values

Arithmetical expression.