# integrateByParts

Integration by parts

## Syntax

``G = integrateByParts(F,du)``

## Description

````G = integrateByParts(F,du)` applies integration by parts to the integrals in `F`, in which the differential `du` is integrated. For more information, see Integration by Parts.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 `integrateByParts` to show the steps of integration by parts.```

example

## Examples

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Create a symbolic expression `F` that is the integral of a product of functions.

```syms u(x) v(x) F = int(u*diff(v))```
```F(x) =  ```

Apply integration by parts to `F`.

`g = integrateByParts(F,diff(u))`
```g =  ```

Apply integration by parts to the integral $\int {\mathit{x}}^{2}\text{\hspace{0.17em}}{\mathit{e}}^{\mathit{x}}\mathit{dx}$.

Define the integral using the `int` function. Show the result without evaluating the integral by setting the `'Hold'` option to `true`.

```syms x F = int(x^2*exp(x),'Hold',true)```
```F =  ```

To show the steps of integration, apply integration by parts to `F` and use `exp(x)` as the differential to be integrated.

`G = integrateByParts(F,exp(x))`
```G =  ```
`H = integrateByParts(G,exp(x))`
```H =  ```

Evaluate the integral in `H` by using the `release` function to ignore the `'Hold'` option.

`F1 = release(H)`
`F1 = $2 {\mathrm{e}}^{x}+{x}^{2} {\mathrm{e}}^{x}-2 x {\mathrm{e}}^{x}$`

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

`F2 = int(x^2*exp(x))`
`F2 = ${\mathrm{e}}^{x} \left({x}^{2}-2 x+2\right)$`

Apply integration by parts to the integral $\int {\mathit{e}}^{\mathit{ax}}\text{\hspace{0.17em}}\mathrm{sin}\left(\mathit{bx}\right)\text{\hspace{0.17em}}\mathit{dx}$.

Define the integral using the `int` function. Show the integral without evaluating it by setting the `'Hold'` option to `true`.

```syms x a b F = int(exp(a*x)*sin(b*x),'Hold',true)```
```F =  ```

To show the steps of integration, apply integration by parts to `F` and use ${\mathit{u}}^{\prime }\left(\mathit{x}\right)={\mathit{e}}^{\mathit{ax}}$ as the differential to be integrated.

`G = integrateByParts(F,exp(a*x))`
```G =  ```

Evaluate the integral in `G` by using the `release` function to ignore the `'Hold'` option.

`F1 = release(G)`
```F1 =  $\frac{{\mathrm{e}}^{a x} \mathrm{sin}\left(b x\right)}{a}-\frac{b {\mathrm{e}}^{a x} \left(a \mathrm{cos}\left(b x\right)+b \mathrm{sin}\left(b x\right)\right)}{a \left({a}^{2}+{b}^{2}\right)}$```

Simplify the result.

`F2 = simplify(F1)`
```F2 =  $-\frac{{\mathrm{e}}^{a x} \left(b \mathrm{cos}\left(b x\right)-a \mathrm{sin}\left(b x\right)\right)}{{a}^{2}+{b}^{2}}$```

## Input Arguments

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

Example: `int(u*diff(v))`

Differential to be integrated, specified as a symbolic variable, expression, or function.

Example: `diff(u)`

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### Integration by Parts

Mathematically, the rule of integration by parts is formally defined for indefinite integrals as

`$\int u\text{'}\left(x\right)\text{\hspace{0.17em}}v\left(x\right)\text{\hspace{0.17em}}dx=u\left(x\right)\text{\hspace{0.17em}}v\left(x\right)-\int u\left(x\right)\text{\hspace{0.17em}}v\text{'}\left(x\right)\text{\hspace{0.17em}}dx$`

and for definite integrals as

`$\underset{a}{\overset{b}{\int }}u\text{'}\left(x\right)\text{\hspace{0.17em}}v\left(x\right)\text{\hspace{0.17em}}dx=u\left(b\right)\text{\hspace{0.17em}}v\left(b\right)-u\left(a\right)\text{\hspace{0.17em}}v\left(a\right)-\underset{a}{\overset{b}{\int }}u\left(x\right)\text{\hspace{0.17em}}v\text{'}\left(x\right)\text{\hspace{0.17em}}dx.$`

## Version History

Introduced in R2019b