Integration by Parts

Integration by Parts 

The formula for integration by parts is:

The integral of 'v' with respect to 'u' is equal to 'uv' less the integral of 'u' with respect to 'v'.

Guidelines for Choosing u and dv

When deciding which part of the integrand should be u and which should be dv, follow the LIATE rule. This is a priority list:

L – Logarithmic functions

I – Inverse trigonometric functions

A – Algebraic functions (polynomials, powers of xxx)

T – Trigonometric functions

E – Exponential functions

How to Apply the LIATE Rule

• Look at the product of functions in your integral.

• The function that appears first in the LIATE list becomes u.

• The remaining part of the integrand becomes dv.

Example

Find:

Here, we have:

• x³  → Algebraic function (A in LIATE)

• log x → Logarithmic function (L in LIATE)

Since L comes before A in the LIATE list, we choose:

u = log x  and dv = x³ dx

We then proceed with the integration by parts formula.