Integration

Standard integrals

(Each plus a constant C.)

Techniques

• Substitution: if the integrand contains g(x) and g′(x), put u = g(x). Example: ∫ 2x eˣ² dx = eˣ² + C.
• Integration by parts: ∫ u dv = uv − ∫ v du. Choose u in the order ILATE (inverse trigonometric, logarithmic, algebraic, trigonometric, exponential). Example: ∫₀¹ x eˣ dx = [x eˣ]₀¹ − ∫₀¹ eˣ dx = e − (e − 1) = 1.
• Partial fractions for rational functions; trigonometric identities such as sin² x = (1 − cos 2x)/2.

Properties of definite integrals

The a − x property solves integrals that look hard. Let I = ∫₀^(π/2) sin x / (sin x + cos x) dx. Replacing x by π/2 − x swaps sin and cos, so I = ∫₀^(π/2) cos x / (cos x + sin x) dx. Adding the two forms: 2I = ∫₀^(π/2) 1 dx = π/2, so I = π/4.

Symmetry: ∫₋₁¹ x³ cos x dx = 0, because x³ cos x is odd.

Areas

The area between y = f(x) and y = g(x) from a to b (with f ≥ g) is ∫ₐᵇ (f(x) − g(x)) dx. Between y = x and y = x², which meet at x = 0 and x = 1: ∫₀¹ (x − x²) dx = 1/2 − 1/3 = 1/6.

When a function changes sign, area and integral differ: ∫₀³ |x − 1| dx splits at x = 1 into 1/2 + 2 = 2.5.

Checking answers numerically

Simpson's rule approximates ∫ₐᵇ f by fitting parabolas through equally spaced points: (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + f(xₙ)] with n even. It is very accurate for smooth functions, and a good way to check a hand computation:

import math

def simpson(f, a, b, n=1000):
    h = (b - a) / n
    total = f(a) + f(b) + sum((4 if i % 2 else 2) * f(a + i * h) for i in range(1, n))
    return total * h / 3

checks = [
    ("x e^x on [0, 1]", lambda x: x * math.exp(x), 0, 1, 1),
    ("sin^2 x on [0, pi/2]", lambda x: math.sin(x) ** 2, 0, math.pi / 2, math.pi / 4),
    ("x sin x on [0, pi]", lambda x: x * math.sin(x), 0, math.pi, math.pi),
    ("sin/(sin+cos) on [0, pi/2]", lambda x: math.sin(x) / (math.sin(x) + math.cos(x)), 0, math.pi / 2, math.pi / 4),
    ("x - x^2 on [0, 1]", lambda x: x - x * x, 0, 1, 1 / 6),
]
for name, f, a, b, exact in checks:
    print(f"{name:28} Simpson {simpson(f, a, b):.6f}   exact {exact:.6f}")

x e^x on [0, 1]              Simpson 1.000000   exact 1.000000
sin^2 x on [0, pi/2]         Simpson 0.785398   exact 0.785398
x sin x on [0, pi]           Simpson 3.141593   exact 3.141593
sin/(sin+cos) on [0, pi/2]   Simpson 0.785398   exact 0.785398
x - x^2 on [0, 1]            Simpson 0.166667   exact 0.166667

Trapezoidal rule: (h/2)[f(x₀) + 2f(x₁) + … + 2f(xₙ₋₁) + f(xₙ)]. Its error is O(h²), compared with O(h⁴) for Simpson's rule; the trapezoidal rule is exact for straight lines, Simpson's rule for polynomials up to degree 3.

Practice questions