# Integration by Parts

# Integration by Parts

## Definition

**Integration by parts**is a method used in calculus to integrate products of two functions.- It is based on the rule of differentiation for the product of two functions, in reverse.

## The Formula

- The integration by parts formula is ∫udv = uv - ∫vdu, where
**u**and**v**are functions of x. - This formula comes directly from the product rule for differentiation.

## Choosing u and dv

- A common mnemonic used to choose the function to differentiate (
**u**) and the function to integrate (**dv**) is ILATE, which stands for Inverse, Logarithmic, Algebraic, Trigonometric and Exponential. - One should choose the function to be
**u**which comes first in this ordering, if possible.

## Implementation Steps

- After deciding on
**u**and**dv**, evaluate**v**by integrating**dv**and**du**by differentiating**u**. - Substitute these into the integration by parts formula: ∫udv = uv - ∫vdu.

## Examples

- The integration by parts formula can simplify integrals such as ∫x cosx dx. Here one would let u = x and dv = cosx dx.
- After differentiating and integrating to find du and v, substitute into the formula to find the integral.

## Significance

**Integration by parts**is an important method in calculus because it allows us to simplify and solve complex integrals, particularly where the integral is the product of an algebraic function (e.g. x) and a trigonometric or exponential function.

## Common Misconceptions

- While deciding what to choose for
**u**and**dv**, remember it does not always have to be in the order of the mnemonic ILATE. - Do not confuse
**integration by parts**with other techniques of integration like**integration by substitution**. The integration by parts formula has a very specific form and is used primarily for the product of two functions.

## Key Points to Remember

- The formula for
**integration by parts**(∫udv = uv - ∫vdu) comes directly from the product rule for differentiation. - A good choice for
**u**and**dv**can simplify the problem a great deal. - Sometimes it’s necessary to apply the formula multiple times or use it together with other techniques such as the method of substitution.
**Integration by parts**is an essential tool in calculus for handling more complex integrals.