Integration by Parts is a method used in calculus to find the integral of a product of two functions. It involves rewriting the integral in a different form, splitting it into two parts, and then applying integration rules to solve for the integral.
In Integration by Parts, we use the acronym "LIATE" to determine which function to integrate and which function to differentiate. This stands for: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential. We choose the first function in the acronym as the one to integrate, and the second function as the one to differentiate.
No, Integration by Parts is most effective for integrals involving products of two functions. It may not be as useful for integrals that do not involve products, or for integrals with more than two functions.
The general formula for Integration by Parts is ∫uv dx = u∫v dx - ∫u' (∫v dx) dx, where u is the first function, v is the second function, and u' is the derivative of u.
Integration by Parts can be used multiple times if necessary, but in most cases, it is effective after two or three repetitions. You can stop using Integration by Parts when the resulting integral becomes simpler and easier to solve.