Integration by Parts is a mathematical technique used in calculus to find the integral of a product of two functions. It involves choosing one function to differentiate and the other to integrate, resulting in a new integral that is hopefully easier to solve.
Integration by Parts is typically used when the integrand (the expression inside the integral) is a product of two functions, and when other integration techniques such as substitution or partial fractions are not effective.
The choice of which function to differentiate and which one to integrate is typically made using the acronym "LIATE". This stands for logarithmic, inverse trigonometric, algebraic, trigonometric, and exponential. In general, the function that comes first in this list should be differentiated, while the one that comes last should be integrated.
One common mistake is to apply the product rule for derivatives to the original integrand. This is incorrect, as Integration by Parts involves applying the product rule for integrals. Another mistake is to forget to include the constant of integration when solving the new integral.
Some tips for making Integration by Parts easier include choosing the function to differentiate wisely, looking for patterns in the new integral, and practicing with a variety of different examples. It is also helpful to check your answer by differentiating it to see if it matches the original integrand.