Welcome back to class! Today we are starting Chapter 7 with a fundamental technique in Calculus 2: Integration by Parts. If you remember the Product Rule from differentiation, this is essentially how we undo it. This method allows us to integrate products of functions that standard substitution can't handle.

The Core Formula

The formula for Integration by Parts is derived directly from the Product Rule. In its most memorable form, using variables $u$ and $v$, it looks like this:

$$\int u \, dv = uv - \int v \, du$$

The goal here is to choose $u$ and $dv$ such that the new integral on the right side, $\int v \, du$, is easier to solve than the original one.

Strategy: How to Choose $u$ (LIATE)

One of the biggest questions students have is, "Which part is $u$ and which part is $dv$?" To solve this, we use the acronym LIATE. You generally want to pick your $u$ value based on which function type appears first in this list:

  • L - Logarithmic (e.g., $\ln x$)
  • I - Inverse Trigonometric (e.g., $\tan^{-1} x$)
  • A - Algebraic (e.g., $x$, $x^2$)
  • T - Trigonometric (e.g., $\sin x$, $\cos x$)
  • E - Exponential (e.g., $e^x$)

Class Examples Breakdown

1. The Classic Example: $\int x \sin x \, dx$

Using LIATE, we have an Algebraic term ($x$) and a Trigonometric term ($\sin x$). Algebraic comes before Trig, so we set:

  • $u = x$ $\rightarrow$ $du = dx$
  • $dv = \sin x \, dx$ $\rightarrow$ $v = -\cos x$

Plugging this into our formula gives us the result: $-x\cos x + \sin x + C$.

2. The Hidden Product: $\int \ln x \, dx$

This looks like a single function, but we treat it as a product of $\ln x$ and $1$. Since Logarithms come first in LIATE:

  • $u = \ln x$
  • $dv = 1 \, dx$

This clever setup allows us to solve an integral that previously seemed impossible!

3. The "Boomerang" Method: $\int e^x \sin x \, dx$

Sometimes, you have to do Integration by Parts twice, and you end up with the same integral you started with on the right side! Don't panic—this is algebraic gold. As shown in the notes, if you have:

$$\int e^x \sin x \, dx = \text{stuff} - \int e^x \sin x \, dx$$

You can simply add the integral to both sides (getting 2 times the integral) and then divide by 2 to solve it.

Important Reminders

  • Check for U-Sub First: Look at the example from the notes: $\int 2x\sqrt{1+x^2} \, dx$. Just because we are learning a new method doesn't mean we forget the old ones! This specific problem is much faster using standard U-Substitution ($u = 1+x^2$). Always check the easy path first.
  • Definite Integrals: When doing definite integrals (like $\int_0^1 \tan^{-1} x \, dx$), remember that your limits of integration apply to the $uv$ term and the new integral.

Integration by Parts takes practice to spot the patterns, but once you master the LIATE rule, it becomes a very systematic process. Keep practicing, and don't forget to record your work!