Integration by Parts 🐸

What is Integration by Parts?

Integration by parts is a powerful technique from calculus used to integrate the products of functions. Derived from the product rule for differentiation, it allows us to transform difficult integrals into simpler ones.

The Formula

The formula for integration by parts is:
∫u dv = uv - ∫v du

How to Choose u and dv

A common mnemonic to help choose u and dv is "LIATE":

1. Logarithmic functions
2. Inverse trigonometric functions
3. Algebraic functions
4. Trigonometric functions
5. Exponential functions

Highest priority to the leftmost function available in your integral.

Example Problem

Let's solve ∫x e^x dx using integration by parts.
Select u = x and dv = e^x dx.
Then, differentiate to get du = dx and integrate to get v = e^x.
Now, apply the formula:

∫x e^x dx = x e^x - ∫e^x dx = x e^x - e^x + C