Welcome back, class! Tonight we are beginning Chapter 7: Techniques of Integration. Specifically, we are tackling Section 7-1: Integration by Parts. This is one of the most powerful tools in your calculus toolkit.

The Big Idea: The Reverse Product Rule

Remember when we learned the Product Rule for differentiation? We know that:

$$ \frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x) $$

Integration by parts is essentially the Product Rule in reverse. It allows us to integrate products of functions that simple substitution can't handle. The formula we will use repeatedly is:

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

Strategy: How to Choose $u$ and $dv$

Success with this technique relies entirely on how you split up your integral. A helpful mnemonic found in the class notes is the LIATE principle. When choosing which part of the integrand to set as $u$, look for functions in this order of precedence:

  • 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., $\sin x$)
  • E - Exponential (e.g., $e^x$)

Example from Class Notes

Let's look at a classic example from the attached notes: $ \int x \sin x \, dx $.

Using LIATE, we have an Algebraic function ($x$) and a Trigonometric function ($\sin x$). Algebraic comes first, so we choose:

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

Plugging this into our formula:

$$ \int x \sin x \, dx = (x)(-\cos x) - \int (-\cos x) \, dx $$$$ = -x \cos x + \int \cos x \, dx $$$$ = -x \cos x + \sin x + C $$

Advanced Cases

In the lecture video and notes, we also cover:

  • Repeated Use: Integrals like $\int x^2 e^x \, dx$ require applying the formula twice.
  • "The Boomerang": Integrals like $\int e^x \sin x \, dx$ loop back on themselves, requiring us to solve for the integral algebraically.
  • Definite Integrals: Don't forget to evaluate your limits, as seen in the $\int_0^1 \tan^{-1} x \, dx$ example!

Please review the PowerPoint and the handwritten PDF notes attached below before our meeting. Bring your questions!

Class Resources:
Zoom Meeting Link | Section 7-1 PowerPoint | Section 7-1 Class Notes (PDF)