Integration Techniques

By SkytrailGroup Mathematics Editorial Team

Substitution

Let u=g(x), du=g'(x)dx: \(\int f(g(x))g'(x)dx=\int f(u)du\).

Integration by Parts

\[\int u\,dv=uv-\int v\,du\]

Partial Fractions

Decompose rational functions into simpler fractions before integrating.

Examples

Example 1. Find \(\int x e^x dx\).

Solution. Parts: u=x, dv=eˣdx → xeˣ−eˣ+C.

Deep Dive: Integration Techniques

This section builds durable understanding of integration techniques in calculus through definition-first reasoning, theorem mapping, and error-checking workflows.

Use a two-pass method: first derive the structure symbolically, then validate with a concrete numerical or geometric test case.

Visual Intuition

Convert algebra into a diagram, graph, or dependency map before solving. Visual-first analysis reduces sign errors and makes assumptions explicit.

Checklist: domain constraints - symmetry - limiting behavior - sanity check at special values.

Practice Set

Practice A. Re-derive one key formula on this page from first principles and annotate each transformation.

Target. Your final line should include assumptions, derivation path, and a quick verification.

Practice B. Build an application scenario using integration techniques and solve it with both symbolic and numeric methods.

Target. Compare outputs and explain any approximation gap.

References & Editorial Notes

  • Stewart, Calculus.
  • Strang, Introduction to Linear Algebra.
  • Apostol, Mathematical Analysis.

Editorial update: Reviewed on 2026-04-14 for notation consistency, conceptual clarity, and exercise quality.