Integrating by Parts

Integrating by Parts

Understanding Integration by Parts

Integration by parts is a method in calculus often used when the integral is the product of two functions, where one can easily be integrated and the other easily differentiated.
The formula used for integration by parts is ∫udv = uv - ∫vdu.
The choice of u and dv is crucial to simplify the integral. Usually, u is chosen as a function that becomes simpler when derived.

Process of Integration by Parts

Familiarise yourself with the method of selecting the function to differentiate (u) and the function to integrate (dv).
Generally u is chosen such that its derivative is simpler than the original function.
The function dv should be easily integrable.
Once you have chosen u and dv, calculate du (derivative of u) and v (integral of dv).
Substitute these into the formula ∫udv = uv - ∫vdu.

Application of Integration by Parts

Practice using this method by applying it to various mathematical questions that involve integration of the product of two functions.
Pay attention to functions that are products of a polynomial and logarithmic or trigonometric functions - these cases often need this method.
Once comfortable, move on to problems where you need to apply the integration by parts method multiple times.

Dealing with Repeated Calculations

When applying integration by parts twice or more, you might end up with the initial integral you started with.
To solve this, you will encounter equations where the integral you are trying to solve is part of the solution - a subtraction technique is used to bring the integral to one side and solve.

Practicing Problems

Aim at mastering the selection process for u and dv - the right choice is half the solution in the method of integration by parts.
Frequently, it is helpful to use the acronym LIATE to decide on the u function: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential.
Solve a range of problems of increasing complexity to get accustomed to different scenarios.