Lesson 13 of 13 · 16 min
Chapter review
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Must-know facts
19 facts
- 01∫f(x) dx = F(x) + C where F′ = f; any two antiderivatives differ by a constant.
- 02∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1; ∫dx/x = log|x| + C.
- 03∫aˣ dx = aˣ/log a + C; ∫eˣ dx = eˣ + C.
- 04∫tan x dx = log|sec x| + C; ∫sec x dx = log|sec x + tan x| + C.
- 05∫cot x dx = log|sin x| + C; ∫cosec x dx = log|cosec x − cot x| + C.
- 06∫f′(x)/f(x) dx = log|f(x)| + C.
- 07∫dx/(x² + a²) = (1/a) tan⁻¹(x/a) + C; ∫dx/√(a² − x²) = sin⁻¹(x/a) + C.
- 08∫dx/(x² − a²) = (1/2a) log|(x − a)/(x + a)| + C.
- 09∫dx/√(x² ± a²) = log|x + √(x² ± a²)| + C.
- 10Improper rational function: divide first, then split into partial fractions.
- 11By parts: ∫u v dx = u∫v dx − ∫(u′∫v dx) dx; ∫log x dx = x log x − x + C.
- 12∫eˣ[f(x) + f′(x)] dx = eˣ f(x) + C.
- 13∫eˣ sin x dx = (eˣ/2)(sin x − cos x) + C.
- 14A(x) = ∫ₐˣ f(t) dt has A′(x) = f(x) when f is continuous.
- 15∫ₐᵇ f(x) dx = F(b) − F(a); no + C in a definite integral.
- 16After a substitution in a definite integral, change the limits too.
- 17∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b − x) dx; ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx.
- 18Odd f on [−a, a] integrates to 0; even f gives 2∫₀ᵃ f dx.
- 19∫₀^(π/2) log sin x dx = −(π/2) log 2.
Common traps
Where marks are lost
Formulas
12 to know
Power rule
∫xⁿ dx = xⁿ⁺¹/(n + 1) + C
n ≠ −1; for n = −1 the answer is log|x| + C.
Exponential
∫aˣ dx = aˣ/log a + C
a > 0, a ≠ 1; for a = e this is eˣ + C.
Tan and sec
∫tan x dx = log|sec x| + C; ∫sec x dx = log|sec x + tan x| + C
By substitution.
Sum of squares
∫dx/(x² + a²) = (1/a) tan⁻¹(x/a) + C
Complete the square first for a general quadratic.
Difference of squares
∫dx/(x² − a²) = (1/2a) log|(x − a)/(x + a)| + C
For a² − x² the fraction inside the log is (a + x)/(a − x).
Root forms
∫dx/√(a² − x²) = sin⁻¹(x/a) + C; ∫dx/√(x² ± a²) = log|x + √(x² ± a²)| + C
a > 0.
Integration by parts
∫u v dx = u∫v dx − ∫(u′ ∫v dx) dx
u is the first function, v the second.
eˣ pattern
∫eˣ[f(x) + f′(x)] dx = eˣ f(x) + C
Spot a function and its derivative.
Root of a² − x²
∫√(a² − x²) dx = (x/2)√(a² − x²) + (a²/2) sin⁻¹(x/a) + C
The x² ± a² forms use a log term instead.
Fundamental theorem
∫ₐᵇ f(x) dx = F(b) − F(a)
F′ = f, f continuous on [a, b].
Reflection property
∫ₐᵇ f(x) dx = ∫ₐᵇ f(a + b − x) dx
With a = 0: ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx.
Even and odd
∫₋ₐᵃ f dx = 2∫₀ᵃ f dx (f even); 0 (f odd)
Check f(−x) first.
Key terms
7 terms
- Antiderivative
- A function F whose derivative is the given f; also called a primitive.
- Indefinite integral
- The family F(x) + C of all antiderivatives of f.
- Integrand
- The function f(x) being integrated.
- Constant of integration
- The arbitrary real number C that shifts one antiderivative into another.
- Proper rational function
- P(x)/Q(x) with the degree of P less than the degree of Q.
- Definite integral
- ∫ₐᵇ f(x) dx, a single number; for f ≥ 0 the area under the curve from a to b.
- Area function
- A(x) = ∫ₐˣ f(t) dt, the area from a fixed a up to a moving x.