Here is a guide on Integration by Parts. It is a tricky Calculus topic at first but it gets easier with practice.
Before continuing, one should be familiar with antiderivatives, the product rule and substitutions with integrals.
- What is Integration by Parts?
- Integration by Parts: The Formula
- The LIATE Memory Aid for Integration by Parts
- Some Examples of Integration by Parts with LIATE
What is Integration by Parts?
Integration by Parts is an integration method for integrating functions like this:
Intergration by Parts: The Formula
The formula for Integration by Parts is:
One could ask what are , , , and ? We will look at the derivation of the formula.
To start, the product rule gives us:
Integrating both sides gives us:
By the Fundamental Theorem of Calculus, the integral of a derivative is the function (integrand) itself. The left side is just as follows:
The substitutions , along with and turn the above line into:
Rearranging the terms in (2) would give the integration by parts formula as given in (1) above.
The LIATE Memory Aid for Integration by Parts
You now know what , , , and are. A natural question would be how do I know which function should be and in the substitution for Integration by Parts? The LIATE principle can help determine what to pick for and . The acronym LIATE stands for:
Top choices for start from the letter L and go down and the top choices for start from the letter E for exponential and go up.
The rationale behind the LIATE principle is that logarithms have no known antiderivative so they are a common choice for the substitution and that the antiderivative of an exponential such as is an exponential.
An alternate acronym is LIPTE where the only difference is that the A for algebraic turns into P for polynomial.
Some Examples of Integration by Parts with LIATE
From before we had:
The algebraic function is and is the exponential function. Our choice for is (algebraic is higher than exponential) and would be . We would then have and .
Substituting the components , , , and into the Integration By Parts formula gives us:
The logarithm would be the choice for and we would have . The derivative of is and the integral of is .
We have an inverse trigonometry function such as and an algebraic/polynomial function such as . The choice for is and we would have . The other components would be and .
The examples above were simple cases. Do be aware that product rule, quotient rule, and chain rule may be needed for determining from .
Multiple integration by parts may be needed at times.