Integration by Parts

Learning Objectives

After reviewing this study guide, you should be able to:

• Derive the integration-by-parts formula from the product rule of differentiation.
• Identify when to use integration by parts versus standard substitution.
• Apply the LIATE mnemonic to correctly select $u$ and $dv$ for indefinite integrals.
• Extend the integration-by-parts formula to evaluate definite integrals and find areas bounded by curves.

---

Key Terms & Glossary

• Integration by Parts: A technique based on the product rule for differentiation that allows us to exchange a complicated integral for a simpler one.
  • Example: Finding the displacement of a vehicle when velocity is a product of time and exponential acceleration (e.g., $\int t e^t dt). We "dismantle" the t through differentiation while "building up" the e^t$ through integration.
• LIATE Rule: An acronym for Logarithmic, Inverse Trigonometric, Algebraic, Trigonometric, and Exponential functions, used to prioritize the choice of $u$.
  • Example: In computing the energy of a system modeled by $\int x \ln(x) dx$, $\ln(x)$ (Logarithmic) comes before $x (Algebraic) in LIATE, so u = \ln(x)$.
• Definite Integral: An integral evaluated between an upper and lower limit, often representing the exact area under a curve.
  • Example: Calculating the exact total square footage of a curved architectural roof between the boundaries $x=0$ and $x=10$.

---

The "Big Idea"

Many students ask, "Is there a product rule for integration?"

The short answer is no. However, Integration by Parts is the mathematical equivalent of reverse-engineering the differentiation product rule. It acknowledges that integrating a direct product of two functions (like $x$ and $\sin x$) defies basic rules. Instead of solving it directly, this technique transforms the difficult integral into a new expression minus a secondary, hopefully much easier, integral.

[!IMPORTANT] The Goal of Integration by Parts You are not immediately finding the final answer. You are exchanging a hard integral $\int u \, dv for an easier integral \int v \, du$.

---

Formula / Concept Box

---

Hierarchical Outline

• 1. The Origin of the Technique
  • Derived by reversing the product rule for differentiation.
  • Solves integrals that look simple but defy $u$-substitution (e.g., $\int x \sin x \, dx$).
• 2. The Core Formula
  • Select a piece to differentiate ($u) and a piece to integrate (dv$).
  • Calculate the derivative ($du) and the integral (v$).
  • Apply: $\int u \, dv = uv - \int v \, du$.
• 3. Choosing $u$ and $dv$ with LIATE
  • Logarithmic (e.g., $\ln x$)
  • Inverse Trig (e.g., $\arctan x$)
  • Algebraic (e.g., $x^2$, $3x$)
  • Trigonometric (e.g., $\sin x$, $\cos x$)
  • Exponential (e.g., $e^x$)
  • Rule of Thumb: Pick the function highest on this list to be $u$.
• 4. Definite Integrals Application
  • Evaluated exactly like the indefinite form, but applying boundaries $[a, b]$ to the $uv$ term.
  • Used to find the exact area of complex bounded regions.

---

Visual Anchors

Flowchart: The Integration by Parts Decision Matrix

TikZ: Visualizing the Area Found by Definite Integration

---

Definition-Example Pairs

• Term: Trial and Error in Selection

  • Definition: The process of attempting to set $u$ and $dv, and restarting if the resulting \int v \, du$ becomes more complicated than the original.
  • Example: Evaluating $\int x e^x \, dx$. Choosing $u = e^x$ yields $\int \frac{1}{2}x^2 e^x \, dx$ (harder!). Restarting and choosing $u = x$ yields $\int e^x \, dx$ (easier!).

• Term: Area Bounded by Curve

  • Definition: The total shaded region sitting directly between a specific function graph and the x-axis, bounded by defined starting and ending points.
  • Example: Finding the physical area of a parabolic archway from the ground up to the ceiling by integrating $\int_0^{10} x \cos(x) \, dx$.

---

Comparison Tables

Choosing $u$ for $\int x \sin x \, dx$

---

Worked Examples

▶Example 1: Indefinite Integration by Parts

Problem: Evaluate $\int x \sin x \, dx$

Step 1: Choose $u$ and $dv$ using LIATE.

• The integrand has an Algebraic function ($x) and a Trigonometric function (\sin x$).
• "A" comes before "T" in LIATE.
• Let $u = x$
• Let $dv = \sin x \, dx$

Step 2: Differentiate to find $du, integrate to find v$.

• $du = 1 \, dx$
• $v = \int \sin x \, dx = -\cos x$

Step 3: Apply the Formula ($uv - \int v \, du$).

• $(x)(-\cos x) - \int (-\cos x) \, dx$
• $-x \cos x + \int \cos x \, dx$

Step 4: Evaluate the final integral.

• $-x \cos x + \sin x + C$

[!TIP] You can check your answer by differentiating $-x \cos x + \sin x + C$. It will simplify exactly back to $x \sin x$!

▶Example 2: Area of a Region (Definite Integral)

Problem: Evaluate the area bounded by the graph from $x=0$ to $x=\frac{\pi}{2}$ for the function $x \sin x$. $Area = \int_0^{\pi/2} x \sin x \, dx$

Step 1: Use the antiderivative found previously. From Example 1, we know the antiderivative is $-x \cos x + \sin x$.

Step 2: Apply the integration by parts for definite integrals. $\left[ -x \cos x + \sin x \right]_0^{\pi/2}$

Step 3: Evaluate at upper and lower boundaries.

• Upper limit (${\pi/2}$): $-({\pi/2})\cos({\pi/2}) + \sin({\pi/2}) = -({\pi/2})(0) + 1 = 1$
• Lower limit (0): $-(0)\cos(0) + \sin(0) = 0 + 0 = 0$

Step 4: Subtract.

• Area $= 1 - 0 = 1$

---

Checkpoint Questions

1. Active Recall: What established rule of differentiation serves as the foundation for the integration-by-parts formula?
2. Active Recall: If an integral contains an exponential function and an algebraic function, which one should be assigned to $u$ and why?
3. Active Recall: When calculating a definite integral using integration by parts, how does the formula differ from the indefinite version?
4. Self-Check: Write down the formula for integration by parts purely from memory. What happens if you make a poor choice for your initial $u$ and $dv$?

[!NOTE] Muddy Points & Cross-Refs If you find yourself repeatedly applying Integration by Parts only to end up back where you started, look up "Tabular Integration" or "The Boomerang Method". These are advanced applications of the exact same principles learned in this chapter!