Integration

Integration accumulates quantities and reverses differentiation. Learn antiderivatives and indefinite integrals, definite integrals as Riemann sums and areas, the fundamental theorem of calculus, standard integrals, substitution, integration by parts, partial fractions, trigonometric integrals and substitutions, and improper integrals.

Distance from velocity, charge from current, energy from power, work from force — whenever a rate is known and the total is wanted, integrate. The fundamental theorem of calculus links integration and differentiation as inverse operations.

An analogy

Integration is like filling a bucket from a tap whose flow changes: the total water is the sum of flow × tiny time intervals.

Area and accumulation

The definite integral is the signed area between the curve and the x-axis (areas below count negative). Always include + C for indefinite integrals.

Techniques

Substitution undoes the chain rule; by parts undoes the product rule (LIATE for choosing u: Log, Inverse trig, Algebraic, Trig, Exponential); partial fractions split rational functions; trig identities and substitutions (x = a sin θ) handle √(a² − x²).

Improper integrals

Infinite limits or unbounded integrands: evaluate as limits. ∫₁^∞ 1/x² dx = 1 converges; ∫₁^∞ 1/x dx diverges.

When no formula exists

Many integrals (e^(−x²), sin x/x) have no elementary antiderivative — use special functions (erf) or numerical integration (trapezoidal, Simpson, Gaussian quadrature — M9).

Formulas

• Fundamental theorem: ∫ₐᵇ f(x) dx = F(b) − F(a), d/dx ∫ₐˣ f(t) dt = f(x)
• Standard integrals: ∫xⁿ dx = xⁿ⁺¹/(n + 1) (n ≠ −1), ∫1/x dx = ln|x|, ∫eˣ = eˣ, ∫sin x = −cos x, ∫cos x = sin x
• Substitution: ∫ f(g(x)) g′(x) dx = ∫ f(u) du
• By parts: ∫ u dv = uv − ∫ v du
• Riemann sum: ∫ₐᵇ f dx = lim_{n→∞} Σ f(xᵢ) Δx

Worked example: Definite integral

Given: ∫₀^π sin x dx

Find: Value.

• F = −cos x
• −cos π + cos 0 = 1 + 1

Answer: 2

Worked example: By parts

Given: ∫ x eˣ dx

Find: Integral.

• u = x, dv = eˣ dx → du = dx, v = eˣ
• x eˣ − ∫ eˣ dx

Answer: eˣ(x − 1) + C

Common mistakes

• Wrong: Forgetting + C. — Right: Indefinite integrals include a constant.
• Wrong: Changing variable but not the limits. — Right: Convert limits when substituting in definite integrals.

Applications

• Charge & energy: q = ∫ i dt, E = ∫ P dt.
• Motion: Distance from velocity.
• Signal processing: RMS, averages, Fourier coefficients.

Key takeaways

• Integration accumulates rates.
• ∫ₐᵇ f = F(b) − F(a).
• Substitution, parts, partial fractions, trig methods.
• Improper integrals as limits.

Going further

• Numerical integration: Trapezoidal, Simpson, Gauss (M9).
• Special functions: erf, Gamma for non-elementary integrals (M15).
• Contour integration: Complex methods for real integrals (M5).