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Tony's Intertwingled Memex • Reference Manual

Integration by Parts

Formulas & Definitions
Core Definition

A rule of integration that reverses the Product Rule of differentiation. For differentiable functions u(x)u(x) and v(x)v(x):

∫u dv=uv−∫v du\int u \, dv = uv - \int v \, du

Concept Explanation

Integration by parts converts a difficult product integral into a simpler one. To select uu effectively, use the LIATE mnemonic priority (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential). You differentiate uu to get dudu, and integrate dvdv to get vv.

Visual / Geometric Intuition

Interactive Desmos Graph

Engineering Applications

Extensively applied in signal analysis to solve Fourier series coefficients, evaluating center of mass integrals, and modeling mechanical dampening parameters in vibration analysis.

Example Problem

Evaluate:

∫xln⁡x dx\int x \ln x \, dx

Solution

Using LIATE, we select u=ln⁡xu = \ln x (logarithmic) and dv=x dxdv = x \, dx (algebraic).

Step 1: Compute derivatives and integrals

  • u=ln⁡x  ⟹  du=1x dxu = \ln x \implies du = \frac{1}{x} \, dx
  • dv=x dx  ⟹  v=12x2dv = x \, dx \implies v = \frac{1}{2} x^2

Step 2: Apply the formula

∫xln⁡x dx=(ln⁡x)(12x2)−∫(12x2)(12x dx)\int x \ln x \, dx = (\ln x)\left(\frac{1}{2} x^2\right) - \int \left(\frac{1}{2} x^2\right)\left(\frac{1}{2x} \, dx\right)
=12x2ln⁡x−12∫x dx= \frac{1}{2} x^2 \ln x - \frac{1}{2} \int x \, dx
=12x2ln⁡x−12(12x2)+C= \frac{1}{2} x^2 \ln x - \frac{1}{2}\left(\frac{1}{2} x^2\right) + C
=12x2ln⁡x−14x2+C= \frac{1}{2} x^2 \ln x - \frac{1}{4} x^2 + C

Connections & References