∇
VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
Visual Overlays & Layers
Chapter 1 Theory

Advanced Integration Techniques (Parts, Trig Sub & Partial Fractions)

Product rule reversal and algebraic decomposition of rational expressions

Visualize how Integration by Parts trades a difficult integral ∫ u dv for an easier one uv - ∫ v du.

1. The Geometry of Integration by Parts

Integration by parts splits an area in the (u, v) plane: the total bounding rectangle of area u(b)v(b) - u(a)v(a) decomposes into the horizontal integral ∫ u dv and the vertical integral ∫ v du.

∫abu dv=[uv]ab−∫abv du\int_a^b u\,dv = \left[ uv \right]_a^b - \int_a^b v\,du

Quick Reference & Cheat Sheet

LIATE Priority Rule
Logarithmic→Inverse Trig→Algebraic→Trigonometric→Exponential\text{Logarithmic} \to \text{Inverse Trig} \to \text{Algebraic} \to \text{Trigonometric} \to \text{Exponential}
Guideline for choosing u such that du simplifies cleanly.