Integrals

Integration as the reverse of differentiation

If d/dx F(x) = f(x), then F is an antiderivative (primitive) of f and we write ∫f(x) dx = F(x) + C. C is the constant of integration: adding any constant does not change the slope, so there is a whole family of answers.

Always check your answer by differentiating it: you must get back the integrand.

Two basic rules: ∫[f ± g] dx = ∫f dx ± ∫g dx and ∫k f dx = k∫f dx.

Standard integrals

• ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C (n ≠ −1); ∫1/x dx = ln|x| + C
• ∫eˣ dx = eˣ + C; ∫aˣ dx = aˣ/ln a + C
• ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C; ∫sec²x dx = tan x + C
• ∫dx/(x² + a²) = (1/a) tan⁻¹(x/a) + C
• ∫dx/(x² − a²) = (1/2a) ln|(x − a)/(x + a)| + C; ∫dx/(a² − x²) = (1/2a) ln|(a + x)/(a − x)| + C
• ∫dx/√(a² − x²) = sin⁻¹(x/a) + C; ∫dx/√(x² ± a²) = ln|x + √(x² ± a²)| + C
• ∫√(a² − x²) dx = (x/2)√(a² − x²) + (a²/2) sin⁻¹(x/a) + C
• ∫√(x² ± a²) dx = (x/2)√(x² ± a²) ± (a²/2) ln|x + √(x² ± a²)| + C

For ∫dx/(ax² + bx + c), complete the square to reach one of these forms.

Integration by substitution

When the integrand has a function and (a multiple of) its derivative, put the inside function = u.

∫2x cos(x²) dx: u = x², du = 2x dx → ∫cos u du = sin u + C = sin(x²) + C.

Useful results: ∫tan x dx = ln|sec x| + C, ∫sec x dx = ln|sec x + tan x| + C, ∫f′(x)/f(x) dx = ln|f(x)| + C.

Integration by partial fractions

For a proper rational function P(x)/Q(x), split it:

• 1/((x − a)(x − b)) = A/(x − a) + B/(x − b)
• repeated factor: A/(x − a) + B/(x − a)²
• quadratic factor that does not split: (Bx + C)/(x² + bx + c)

Find A, B by putting convenient x values, then integrate each piece as a log. If the degree on top is not smaller, divide first.

Integration by parts

∫u dv = uv − ∫v du, or ∫f·g dx = f∫g dx − ∫[f′ ∫g dx] dx.

Choose u (the first function) by ILATE. ∫x eˣ dx: u = x, dv = eˣ dx → x eˣ − ∫eˣ dx = eˣ(x − 1) + C.

Special result: ∫eˣ[f(x) + f′(x)] dx = eˣ f(x) + C.

Definite integral and the fundamental theorem

∫ₐᵇ f(x) dx is the limit of the sum of thin rectangle areas under y = f(x) from a to b (area above the x-axis counts +, below counts −).

Fundamental theorem of calculus: (1) A(x) = ∫ₐˣ f(t) dt has A′(x) = f(x). (2) If F′ = f, then ∫ₐᵇ f(x) dx = F(b) − F(a). No +C is needed.

With substitution in a definite integral, change the limits too.

Properties and evaluation of definite integrals

1. ∫ₐᵇ f(x) dx = ∫ₐᵇ f(t) dt (the letter does not matter).
2. ∫ₐᵇ f = −∫ from b to a of f; ∫ₐᵃ f = 0.
3. ∫ from a to b = ∫ from a to c + ∫ from c to b (split at c, useful for |x| type functions).
4. ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b − x) dx.
5. ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx.
6. ∫₀²ᵃ f(x) dx = 2∫₀ᵃ f(x) dx if f(2a − x) = f(x), and 0 if f(2a − x) = −f(x).
7. ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx if f is even; 0 if f is odd.

Classic: I = ∫₀^(π/2) sin x/(sin x + cos x) dx. By property 5, I also = ∫ cos x/(cos x + sin x) dx. Add: 2I = ∫₀^(π/2) 1 dx = π/2, so I = π/4.

Key formulas and definitions

• ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C, ∫1/x dx = ln|x| + C
• ∫eˣ dx = eˣ + C, ∫sin x dx = −cos x + C, ∫cos x dx = sin x + C
• ∫dx/(x² + a²) = (1/a) tan⁻¹(x/a) + C
• By parts: ∫u dv = uv − ∫v du
• ∫ₐᵇ f(x) dx = F(b) − F(a)
• ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx

Worked examples

Common mistakes

• Forgetting + C in indefinite integrals (it is not needed in definite ones).
• Using ∫xⁿ dx = xⁿ⁺¹/(n + 1) for n = −1. ∫1/x dx = ln|x| + C.
• After substitution in a definite integral, using the old limits for the new variable.
• Picking the wrong u in by parts (e.g. u = eˣ in ∫x eˣ dx), which makes the new integral harder.