# Integration By Parts

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.

Topics

What is Integration by Parts?

Integration by Parts is an integration method for integrating functions like this:

Here is a polynomial and is an exponential function. Another example is:

The natural question is how do I integrate these things!? We have to use a technique called Integration By Parts.

Intergration by Parts: The Formula

The formula for Integration by Parts is:

(1)

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:

(2)

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

Example One

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:

Source: http://quicklatex.com/cache3/86/ql_72b65e5e9d6688cc2c58c1d6939f4c86_l3.png

Example Two

The logarithm would be the choice for and we would have . The derivative of is and the integral of is .

Example Three

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 .

Notes

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.