# 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.