Integration by Parts

Definition

Integration by parts is a technique for evaluating integrals of products by reversing the Product Rule for differentiation. It is defined by the formula: $\int u \, dv = uv - \int v \, du$

• How to read: “The integral of u with respect to v is equal to u times v minus the integral of v with respect to u.”
• Meaning: Transfers the differentiation burden from $u$ to $v$: differentiate $u$ (getting $du$) and integrate $dv$ (getting $v$), trading one integral for another that is hopefully simpler.

Why It Matters

Many of nature’s laws involve products (e.g., $x e^x$ or $x \sin x$). Integration by parts is the essential tool for unpacking these relationships, allowing us to find totals and accumulations in physical systems where variables are intertwined.

Core Concepts

• Product Rule Reversal: The formula is derived directly from $d(uv) = u \, dv + v \, du$.

  • How to read: “The differential of u times v is equal to u times the differential of v plus v times the differential of u.”
  • Meaning: Rearranging this differential identity yields the integration-by-parts formula—it’s the integral form of the product rule.

• Strategic Choice: The goal is to choose $u$ such that it simplifies when differentiated ($du$) and $dv$ such that it is easily integrable ($v$).

• Tabular Integration: A streamlined method for cases where $u$ is a polynomial that eventually differentiates to zero.

Connected Concepts

• The LIATE Rule
• Integration of Rational Functions by Partial Fractions
• Multiply by Zero