Integration by Parts

Integration by parts is a technique used to find the integral of a product of functions. It is based on the product rule of differentiation and is particularly useful when direct integration is difficult or impossible.

The formula for integration by parts is:

This is often written in the more compact form:

Where:

• is the first function you choose
• is the differential of the second function
• is the integral of
• is the differential of

Integration by parts is particularly useful for integrals of the following forms:

• Products of algebraic and transcendental functions:
• Products of logarithmic and algebraic functions:
• Products involving inverse trigonometric functions:

The key to successfully applying integration by parts is choosing the right functions for and . A helpful mnemonic for choosing is "LIATE":

• L: Logarithmic functions (ln(x), log₁₀(x))
• I: Inverse trigonometric functions (arcsin(x), arctan(x))
• A: Algebraic functions (x, x², √x)
• T: Trigonometric functions (sin(x), cos(x))
• E: Exponential functions (eˣ, aˣ)

Functions higher in this list are generally better choices for , while functions lower in the list are better for .

For integrals requiring multiple applications of integration by parts, the tabular method (also known as the "DI" method) provides a more organized approach.

To use the tabular method:

1. Create a table with two columns labeled D (derivatives) and I (integrals)
2. In the D column, write the first function (u) and its successive derivatives until you reach 0
3. In the I column, write the integral of the second function (v) and its successive integrals
4. Multiply entries diagonally, alternating signs (+ - + - ...)
5. Sum the products to get the final result

Example: Let's evaluate using the tabular method:

Now we multiply diagonally with alternating signs:

This gives us the same result as the repeated application of integration by parts, but in a more systematic and less error-prone way.