Integration by Parts 🐸

The Magic of Integration by Parts

Integration by parts is a magical technique from calculus used to integrate products of functions. It's like the product rule for differentiation, but in reverse. 🌟

The formula for integration by parts is:

∫ u dv = uv - ∫ v du

Where:

• u is a function of x
• dv is the differential of another function of x

Choosing strategically for 'u' and 'dv' can turn a thorny integral into a solvable one. 🔍

Example

Let's say we need to integrate x e^x. We'll set:

• u = x
• dv = e^x dx

Then, du = dx and v = e^x.

Applying the formula:

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

Resulting in a joyful simplified solution! 🎉