Integration as anti-differentiation
If d/dx F(x) = f(x), then F is an antiderivative (or primitive) of f, and we write
∫ f(x) dx = F(x) + C
where C is the constant of integration. Since the derivative of a constant is zero, every function has infinitely many antiderivatives differing by constants.
Standard integrals
∫ f(x) dx | Result |
|---|---|
∫ xⁿ dx (n ≠ −1) | xⁿ⁺¹/(n + 1) + C |
∫ (1/x) dx | log ∣x∣ + C |
∫ eˣ dx | eˣ + C |
∫ aˣ dx | aˣ/log a + C |
∫ sin x dx | −cos x + C |
∫ cos x dx | sin x + C |
∫ sec² x dx | tan x + C |
∫ cosec² x dx | −cot x + C |
∫ sec x tan x dx | sec x + C |
∫ 1/√(1 − x²) dx | sin⁻¹ x + C |
∫ 1/(1 + x²) dx | tan⁻¹ x + C |
Integration is linear: ∫ [k f(x) + g(x)] dx = k ∫ f(x) dx + ∫ g(x) dx.
Example. ∫ (3x² + 4 cos x − 2/x) dx = x³ + 4 sin x − 2 log |x| + C.
Integration by substitution
If the integrand contains a function and (a multiple of) its derivative, substitute t for that function.
Example. ∫ 2x cos(x²) dx. Let t = x², so dt = 2x dx. The integral becomes ∫ cos t dt = sin t + C = sin(x²) + C.
Example. ∫ tan x dx = ∫ sin x/cos x dx. Let t = cos x, dt = −sin x dx: −∫ dt/t = −log |cos x| + C = log |sec x| + C.
Useful results: ∫ f′(x)/f(x) dx = log |f(x)| + C, and ∫ sec x dx = log |sec x + tan x| + C.
Special integrals
Integral | Result |
|---|---|
∫ dx/(x² − a²) | (1/2a) log ∣(x − a)/(x + a)∣ + C |
∫ dx/(a² − x²) | (1/2a) log ∣(a + x)/(a − x)∣ + C |
∫ dx/(x² + a²) | (1/a) tan⁻¹(x/a) + C |
∫ dx/√(a² − x²) | sin⁻¹(x/a) + C |
∫ dx/√(x² + a²) | log ∣x + √(x² + a²)∣ + C |
For quadratics like ax² + bx + c in a denominator, complete the square to reach one of these forms.
Partial fractions
A rational function P(x)/Q(x), with the degree of P less than that of Q, can be split into simpler fractions.
Denominator factor | Partial fraction |
|---|---|
(x − a)(x − b) | A/(x − a) + B/(x − b) |
(x − a)² | A/(x − a) + B/(x − a)² |
(x − a)(x² + bx + c), quadratic irreducible | A/(x − a) + (Bx + C)/(x² + bx + c) |
Example. ∫ dx/((x + 1)(x + 2)). Write 1/((x + 1)(x + 2)) = 1/(x + 1) − 1/(x + 2). Then the integral is log |x + 1| − log |x + 2| + C = log |(x + 1)/(x + 2)| + C.
Integration by parts
∫ u v dx = u ∫ v dx − ∫ [u′ ∫ v dx] dx
Choose u as the function that becomes simpler when differentiated. The ILATE order is a good guide for choosing u: Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential.
Example. ∫ x eˣ dx. Take u = x: x eˣ − ∫ eˣ dx = eˣ(x − 1) + C.
Example. ∫ log x dx. Write it as ∫ (log x)(1) dx: x log x − ∫ (1/x)(x) dx = x log x − x + C.
A useful result: ∫ eˣ [f(x) + f′(x)] dx = eˣ f(x) + C. For example, ∫ eˣ (sin x + cos x) dx = eˣ sin x + C.
Definite integrals
The definite integral ∫ₐᵇ f(x) dx is the area under the curve y = f(x) between x = a and x = b (counting area below the x-axis as negative). It can be defined as the limit of a sum of thin rectangles.
Fundamental theorem of calculus. If F is any antiderivative of f, then
∫ₐᵇ f(x) dx = F(b) − F(a)
The constant C cancels out, so it is not needed.
Example. ∫₀² x² dx = [x³/3]₀² = 8/3 − 0 = 8/3.
Example. ∫₀^(π/2) sin x dx = [−cos x]₀^(π/2) = 0 − (−1) = 1.
When substituting in a definite integral, change the limits too.
Properties of definite integrals
∫ₐᵇ f(x) dx = −∫ᵦₐ f(x) dx, and ∫ₐᵃ f(x) dx = 0.
The integral from a to b equals the integral from a to c plus the integral from c to b, for any point c (split the range).
∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b − x) dx
∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx
∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx if f is even (f(−x) = f(x)), and 0 if f is odd (f(−x) = −f(x)).
Example. ∫₋₁¹ x³ dx = 0, since x³ is odd.
Example (property 4). I = ∫₀^(π/2) sin x ⁄ (sin x + cos x) dx. Replacing x by π/2 − x gives 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.
Summary
∫ f(x) dx = F(x) + C where F′ = f.
Methods: substitution, partial fractions, by parts (ILATE), special forms.
∫ₐᵇ f(x) dx = F(b) − F(a).
∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx; odd functions integrate to 0 over [−a, a].