Integrals
The reverse of differentiation โ finding areas, volumes, and accumulated quantities
In this chapter
7.1 Introduction
Differential calculus deals with rates of change. Integral calculus is its reverse โ it finds the function whose derivative is known, and computes accumulated quantities like area under curves. Together, they form the two pillars of calculus.
7.2 Integration as Inverse of Differentiation
A function F(x) is called an antiderivative (or primitive) of f(x) if F'(x) = f(x).
The indefinite integral is written as:
โซf(x) dx = F(x) + C
where C is the constant of integration โ an arbitrary constant added because differentiation eliminates constants.
7.3 Methods of Integration
Substitution
Put t = g(x) when derivative of inner function is present. Simplifies the integral.
Partial Fractions
Split rational functions P(x)/Q(x) into simpler fractions when deg(P) < deg(Q).
By Parts
โซuยทv dx = uโซv dx โ โซ(u'โซv dx)dx. Use LIATE rule to choose u.
Special Forms
Memorize standard integrals for 1/(xยฒยฑaยฒ), 1/โ(xยฒยฑaยฒ), etc.
7.4 Integrals of Some Particular Functions
For integrals involving axยฒ + bx + c in the denominator:
1. Factor out 'a' from the first two terms
2. Complete the square inside the parentheses
3. Substitute to reduce to standard forms like โซdx/(xยฒยฑaยฒ) or โซdx/โ(xยฒยฑaยฒ)
7.5 Integration by Partial Fractions
For rational functions P(x)/Q(x) where deg(P) < deg(Q), decompose into simpler fractions:
Repeated linear: P(x)/(xโa)ยฒ = A/(xโa) + B/(xโa)ยฒ
Irreducible quadratic: P(x)/[(xโa)(xยฒ+bx+c)] = A/(xโa) + (Bx+C)/(xยฒ+bx+c)
Step 1: If deg(P) โฅ deg(Q), divide first
Step 2: Factor Q(x) completely
Step 3: Write partial fraction form
Step 4: Clear denominators and equate coefficients
Step 5: Solve for A, B, C...
Step 6: Integrate each simple fraction
7.6 Integration by Parts
Pick u as the function that comes first in:
Logarithmic โ Inverse trig โ Algebraic โ Trigonometric โ Exponential
The one earlier in this list becomes u; the other becomes v.
7.7 Definite Integral
A definite integral has limits โ it computes the net signed area between the curve and x-axis from a to b.
7.8 Fundamental Theorem of Calculus
FTC Part 1
d/dx [โซโหฃ f(t) dt] = f(x). Differentiating an integral returns the original function.
FTC Part 2
โซโแต f(x) dx = F(b) โ F(a). Evaluate at endpoints and subtract.
7.9 Evaluation by Substitution
For definite integrals, when substituting t = g(x), also change the limits: when x = a, t = g(a); when x = b, t = g(b). This avoids back-substitution.
7.10 Properties of Definite Integrals
โซโแต f(x) dx = โซโแต f(a + b โ x) dx
Replace every x with (a + b โ x). This is the single most powerful trick for definite integrals.
7.11 Summary
โข Integration is the reverse of differentiation โ โซf(x)dx = F(x) + C where F' = f
โข Three methods: substitution, partial fractions, integration by parts
โข LIATE rule for choosing u in integration by parts
โข Standard integrals: memorize โซdx/(xยฒยฑaยฒ), โซdx/โ(xยฒยฑaยฒ), etc.
โข Completing the square reduces quadratics to standard forms
โข FTC Part 2: โซโแต f(x)dx = F(b) โ F(a)
โข King's Property: โซโแต f(x)dx = โซโแต f(a+bโx)dx
โข Signed area: area above x-axis is positive, below is negative