Integration by parts is a method used to solve integrals that involve the product of two functions. It is based on the product rule of differentiation and involves rewriting the integral in a form that allows for easier integration.
Integration by parts should be used when the integral involves a product of functions where one function is difficult to integrate and the other function is easy to differentiate. This method can also be used when the integral involves a function multiplied by a variable raised to a power.
The formula for integration by parts is ∫u dv = uv - ∫v du, where u and v are functions of x and dv and du are their respective derivatives. This formula is based on the product rule of differentiation.
When using integration by parts, the function u should be chosen as the one that will become simpler when differentiated, and the function dv should be chosen as the one that can be easily integrated. A common method for choosing u is the acronym LIATE, which stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential. The function that falls first in this list should be chosen as u.
Yes, integration by parts can be used to solve definite integrals. After applying the formula and simplifying, the resulting integral can be evaluated at the limits to find the definite integral.