changeIntegrationVariable

Integration by substitution

Syntax

``G = changeIntegrationVariable(F,old,new)``

Description

example

````G = changeIntegrationVariable(F,old,new)` applies integration by substitution to the integrals in `F`, in which `old` is replaced by `new`. `old` must depend on the previous integration variable of the integrals in `F` and `new` must depend on the new integration variable. For more information, see Integration by Substitution.When specifying the integrals in `F`, you can return the unevaluated form of the integrals by using the `int` function with the `'Hold'` option set to `true`. You can then use `changeIntegrationVariable` to show the steps of integration by substitution.```

Examples

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Apply a change of variable to the definite integral ${\int }_{\mathit{a}}^{\mathit{b}}\mathit{f}\left(\mathit{x}+\mathit{c}\right)\text{\hspace{0.17em}}\mathit{dx}$.

Define the integral.

```syms f(x) y a b c F = int(f(x+c),a,b)```
```F =  ${\int }_{a}^{b}f\left(c+x\right)\mathrm{d}x$```

Change the variable $\mathit{x}+\mathit{c}$ in the integral to $\mathit{y}$.

`G = changeIntegrationVariable(F,x+c,y)`
```G =  ${\int }_{a+c}^{b+c}f\left(y\right)\mathrm{d}y$```

Find the integral of $\int \mathrm{cos}\left(\mathrm{log}\left(\mathit{x}\right)\right)\mathit{dx}$ using integration by substitution.

Define the integral without evaluating it by setting the `'Hold'` option to `true`.

```syms x t F = int(cos(log(x)),'Hold',true)```
```F =  $\int \mathrm{cos}\left(\mathrm{log}\left(x\right)\right)\mathrm{d}x$```

Substitute the expression `log(x)` with `t`.

`G = changeIntegrationVariable(F,log(x),t) `
```G =  $\int {\mathrm{e}}^{t} \mathrm{cos}\left(t\right)\mathrm{d}t$```

To evaluate the integral in `G`, use the `release` function to ignore the `'Hold'` option.

`H = release(G)`
```H =  $\frac{{\mathrm{e}}^{t} \left(\mathrm{cos}\left(t\right)+\mathrm{sin}\left(t\right)\right)}{2}$```

Restore `log(x)` in place of `t`.

`H = simplify(subs(H,t,log(x)))`
```H =  $\frac{\sqrt{2} x \mathrm{sin}\left(\frac{\pi }{4}+\mathrm{log}\left(x\right)\right)}{2}$```

Compare the result to the integration result returned by `int` without setting the `'Hold'` option to `true`.

`Fcalc = int(cos(log(x)))`
```Fcalc =  $\frac{\sqrt{2} x \mathrm{sin}\left(\frac{\pi }{4}+\mathrm{log}\left(x\right)\right)}{2}$```

Find the closed-form solution of the integral $\int \mathit{x}\text{\hspace{0.17em}}\mathrm{tan}\left(\mathrm{log}\left(\mathit{x}\right)\right)\mathit{dx}$.

Define the integral using the `int` function.

```syms x F = int(x*tan(log(x)),x)```
```F =  $\int x \mathrm{tan}\left(\mathrm{log}\left(x\right)\right)\mathrm{d}x$```

The `int` function cannot find the closed-form solution of the integral.

Substitute the expression `log(x)` with `t`. Apply integration by substitution.

```syms t G = changeIntegrationVariable(F,log(x),t)```
```G =  ```

The closed-form solution is expressed in terms of hypergeometric functions. For more details on hypergeometric functions, see `hypergeom`.

Compute the integral ${\int }_{0}^{1}{\mathit{e}}^{\sqrt{\mathrm{sin}\left(\mathit{x}\right)}}\mathit{dx}$ numerically with high precision.

Define the integral ${\int }_{0}^{1}{\mathit{e}}^{\sqrt{\mathrm{sin}\left(\mathit{x}\right)}}\mathit{dx}$. A closed-form solution to the integral does not exist.

```syms x F = int(exp(sqrt(sin(x))),x,0,1)```
```F =  ${\int }_{0}^{1}{\mathrm{e}}^{\sqrt{\mathrm{sin}\left(x\right)}}\mathrm{d}x$```

You can use `vpa` to compute the integral numerically to 10 significant digits.

`F10 = vpa(F,10)`
`F10 = $1.944268879$`

Alternatively, you can use the `vpaintegral` function and specify the relative error tolerance.

`Fvpa = vpaintegral(exp(sqrt(sin(x))),x,0,1,'RelTol',1e-10)`
`Fvpa = $1.944268879$`

The `vpa` function cannot find the numerical integration to 70 significant digits, and it returns the unevaluated integral in the form of `vpaintegral`.

`F70 = vpa(F,70)`
`F70 = $\text{vpaintegral}\left({\mathrm{e}}^{\sqrt{\mathrm{sin}\left(x\right)}},x,3.614058973481922839993540324829136186551779737228174541959730561814383e-71,1\right)+3.614058973481922839993540324829136201036215880733963159636656251055722e-71$`

To find the numerical integration with high precision, you can perform a change of variable. Substitute the expression $\sqrt{\mathrm{sin}\left(\mathit{x}\right)}$ with $\mathit{t}$. Compute the integral numerically to 70 significant digits.

```syms t; G = changeIntegrationVariable(F,sqrt(sin(x)),t)```
```G =  ${\int }_{0}^{\sqrt{\mathrm{sin}\left(1\right)}}\frac{2 t {\mathrm{e}}^{t}}{\sqrt{1-{t}^{4}}}\mathrm{d}t$```
`G70 = vpa(G,70)`
`G70 = $1.944268879138581167466225761060083173280747314051712224507065962575967$`

Input Arguments

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Expression containing integrals, specified as a symbolic expression, function, vector, or matrix.

Subexpression to be substituted, specified as a symbolic scalar variable, expression, or function. `old` must depend on the previous integration variable of the integrals in `F`.

New subexpression, specified as a symbolic scalar variable, expression, or function. `new` must depend on the new integration variable.

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Integration by Substitution

Mathematically, the substitution rule is formally defined for indefinite integrals as

`$\int f\left(g\left(x\right)\right)\text{\hspace{0.17em}}g\text{'}\left(x\right)\text{\hspace{0.17em}}dx={\left(\int f\left(t\right)\text{\hspace{0.17em}}dt\right)|}_{t=g\left(x\right)}$`

and for definite integrals as

`$\underset{a}{\overset{b}{\int }}f\left(g\left(x\right)\right)\text{\hspace{0.17em}}g\text{'}\left(x\right)\text{\hspace{0.17em}}dx=\underset{g\left(a\right)}{\overset{g\left(b\right)}{\int }}f\left(t\right)\text{\hspace{0.17em}}dt.$`