Unit 3: Fundamentals of integral calculus

I. Orientation — Integration as Accumulation and Antidifferentiation

Integral calculus studies accumulation and the recovery of a function from its rate of change. Its governing principle is the Fundamental Theorem of Calculus: differentiation and integration are inverse processes under suitable continuity conditions.

A. Defining Principles and Conventions

The essential ideas establish the notation and assumptions used throughout the unit.

• Antiderivative: A function (F) is an antiderivative of (f) on an interval if (F'(x)=f(x)).

∫ f(x) dx = F(x) + C

• (x) is the variable of integration.
• (f(x)) is the integrand.
• (dx) identifies the integration variable.
• (C) is an arbitrary constant because the derivative of any constant is zero.

• Indefinite integral: The expression (\int f(x)\,dx) represents the family of all antiderivatives of (f), not a single numerical value.

• Definite integral: If (f) is continuous on ([a,b]), then

∫[a to b] f(x) dx = F(b) - F(a)

• (a) and (b) are the lower and upper limits.
• (F'(x)=f(x)).
• The result is a number representing signed accumulation.

• Geometric interpretation: (\int_a^b f(x)\,dx) gives signed area: regions above the (x)-axis contribute positively, while regions below it contribute negatively.

• Engineering interpretation: Integration converts a rate into a total. For example, if velocity is (v(t)) metres per second, displacement over ([t_1,t2]) is (\int{t_1}^{t_2}v(t)\,dt) metres.

• Existence assumption: Continuous functions are integrable on closed bounded intervals. Some discontinuous functions are also integrable, but continuity is the standard sufficient condition used here.

II. General Rules of Integration — Standard Antiderivatives and Linearity

A. General Rules of Integration

General rules of integration reverse familiar differentiation formulas and allow complicated integrands to be separated into simpler terms.

• Constant rule: For a constant (k),

∫ k dx = kx + C

• Power rule: For real (n\neq -1),

∫ x^n dx = x^(n+1)/(n+1) + C

• The restriction (n\neq -1) is necessary because division by (n+1) would otherwise be undefined.

• Logarithmic rule: The excluded power (n=-1) has the antiderivative

∫ (1/x) dx = ln|x| + C,  x ≠ 0

• Constant-multiple rule: If (k) is independent of (x),

∫ kf(x) dx = k∫ f(x) dx

• Sum-and-difference rule:

∫[f(x) ± g(x)] dx = ∫f(x) dx ± ∫g(x) dx

• (f) and (g) are integrable functions on the relevant interval.

• Exponential rules:

∫ e^x dx = e^x + C
∫ a^x dx = a^x/ln(a) + C

• The base satisfies (a>0) and (a\neq1).

• Basic trigonometric rules:

∫ sin(x) dx = -cos(x) + C
∫ cos(x) dx = sin(x) + C
∫ sec²(x) dx = tan(x) + C
∫ 1/(1+x²) dx = arctan(x) + C

• Worked example: Integrate (3x^2-4x+5) term by term.

∫(3x² - 4x + 5) dx
= x³ - 2x² + 5x + C

• Differentiating the result produces (3x^2-4x+5), verifying the integration.

B. Applications and Limitations

These rules are most effective when the integrand already resembles standard derivative patterns.

• Application: Polynomial force, velocity, current, and flow-rate models can be integrated term by term to obtain work, displacement, charge, and volume.

• Limitation: The power rule does not directly handle products such as (xe^x), compositions such as (2x\cos(x^2)), or rational functions such as (1/(x^2-1)); these require other methods.

• Verification: An indefinite integral should be checked by differentiation, including the chain-rule factors that are easily omitted.

III. Integration by Substitution — Reversing the Chain Rule

A. Integration by Substitution

Integration by substitution simplifies a composite integrand by replacing an inner expression with a new variable.

• Principle: If (u=g(x)), then (du=g'(x)\,dx), giving

∫ f(g(x))g'(x) dx = ∫ f(u) du

• (g) is differentiable.
• (u) is the substituted variable.
• (f) is integrable over the transformed domain.

• Procedure:

  • Choose (u) as an inner expression whose derivative also appears in the integrand.
  • Compute (du) and replace all relevant occurrences of (x) and (dx).
  • Integrate with respect to (u).
  • For an indefinite integral, substitute (u=g(x)) back into the answer.

• Definite-integral convention: Either return to (x) before evaluating or transform the limits. If (x=a) gives (u=g(a)) and (x=b) gives (u=g(b)), then

∫[a to b] f(g(x))g'(x) dx
= ∫[g(a) to g(b)] f(u) du

• Worked example: Evaluate (\int 2x\cos(x^2)\,dx).

Let u = x², so du = 2x dx.

∫ 2x cos(x²) dx
= ∫ cos(u) du
= sin(u) + C
= sin(x²) + C

• Recognition clue: Useful patterns include (g'(x)[g(x)]^n), (g'(x)e^{g(x)}), (g'(x)/g(x)), and (g'(x)\cos(g(x))).

B. Applications and Limitations

Substitution is appropriate when the integrand contains a composition together with the derivative, or a constant multiple of the derivative, of its inner function.

• Application: The method handles nonlinear variable transformations common in probability densities, changing coordinate systems, and engineering models with exponential or trigonometric responses.

• Limitation: A poor substitution may leave both (x) and (u) in the transformed integral. Every factor must be rewritten consistently in one variable.

• Absolute value: The pattern (\int g'(x)/g(x)\,dx) yields (\ln|g(x)|+C), provided (g(x)\neq0).

IV. Integration by Parts — Reversing the Product Rule

A. Integration by Parts

Integration by parts transforms the integral of a product by transferring differentiation from one factor to another.

• Derivation: Starting from the product rule,

d(uv)/dx = u(dv/dx) + v(du/dx)

integration gives

∫ u dv = uv - ∫ v du

• (u) and (v) are differentiable functions.
• (du=u'(x)\,dx) and (dv=v'(x)\,dx).

• Choice of factors: Select (u) so that differentiating it simplifies the expression, and select (dv) so that it can be integrated readily.

• Common priority guide: Logarithmic and inverse-trigonometric functions are often chosen as (u), followed by algebraic, trigonometric, and exponential functions. This is guidance rather than a theorem.

• Worked example: Integrate (xe^x).

Choose u = x and dv = e^x dx.
Then du = dx and v = e^x.

∫ xe^x dx
= xe^x - ∫ e^x dx
= xe^x - e^x + C
= e^x(x - 1) + C

• Repeated application: Products such as (x^2e^x) require integration by parts more than once because differentiating the polynomial repeatedly eventually reduces it to zero.

B. Applications and Limitations

Integration by parts is especially useful for products in which one factor becomes simpler when differentiated.

• Application: Typical forms include (x^n e^x), (x^n\sin x), (\ln x), and inverse-trigonometric functions, with an implicit factor of (1) where necessary.

• Definite form:

∫[a to b] u dv = [uv] from a to b - ∫[a to b] v du

• Boundary values must be applied to the entire product (uv).

• Limitation: An unsuitable choice of (u) and (dv) can produce a harder integral or return the original expression without useful simplification.

V. Integration by Partial Fraction — Decomposing Rational Functions

A. Integration by Partial Fraction

Integration by partial fraction rewrites a proper rational function as a sum of simpler fractions with standard antiderivatives.

• Precondition: For (P(x)/Q(x)), the degree of polynomial (P) must be less than the degree of polynomial (Q). Otherwise, polynomial division is performed first.

• Distinct linear factors: If (Q(x)=(x-a)(x-b)), then

P(x)/Q(x) = A/(x-a) + B/(x-b)

• (A) and (B) are constants determined by equating coefficients or substituting convenient values of (x).

• Repeated linear factors: A factor ((x-a)^m) requires every power through (m):

A₁/(x-a) + A₂/(x-a)² + ... + Aₘ/(x-a)^m

• (m) is a positive integer.

• Irreducible quadratic factors: A factor (x^2+px+q) that has no real linear factors requires a linear numerator:

(Ax+B)/(x²+px+q)

• (A,B,p,) and (q) are real constants.

• Worked example: Evaluate (\int 1/(x^2-1)\,dx).

1/(x²-1) = 1/[(x-1)(x+1)]
          = 1/[2(x-1)] - 1/[2(x+1)]

∫ 1/(x²-1) dx
= (1/2)ln|x-1| - (1/2)ln|x+1| + C

B. Applications and Limitations

Partial fractions convert rational models into logarithmic, inverse-trigonometric, and power-form integrals.

• Application: The method appears in transform inversion, circuit analysis, control systems, and solutions of separable differential equations.

• Limitation: It applies directly only to rational functions and depends on factoring the denominator over the chosen number system.

• Domain condition: Values that make the original denominator zero remain excluded, even if algebraic simplification appears to remove a factor.

VI. Properties of Definite Integrals — Bounds, Symmetry, and Linearity

A. Properties of Definite Integrals

Properties of definite integrals allow values to be transformed or related without finding a new antiderivative each time.

• Identical limits:

∫[a to a] f(x) dx = 0

• Reversed limits:

∫[a to b] f(x) dx = -∫[b to a] f(x) dx

• Interval addition: For any intermediate point (c),

∫[a to b] f(x) dx
= ∫[a to c] f(x) dx + ∫[c to b] f(x) dx

• Linearity: For constants (\alpha) and (\beta),

∫[a to b] [αf(x)+βg(x)] dx
= α∫[a to b] f(x) dx + β∫[a to b] g(x) dx

• Comparison property: If (f(x)\leq g(x)) throughout ([a,b]), then

∫[a to b] f(x) dx ≤ ∫[a to b] g(x) dx

• Bounds: If (m\leq f(x)\leq M) on ([a,b]), then

m(b-a) ≤ ∫[a to b] f(x) dx ≤ M(b-a)

• (m) and (M) are lower and upper bounds of (f).

• Symmetry on ([-a,a]):
  1. If (f(-x)=f(x)), then (f) is even and

∫[-a to a] f(x) dx = 2∫[0 to a] f(x) dx

2. If (f(-x)=-f(x)), then (f) is odd and

∫[-a to a] f(x) dx = 0

• Reflection property:

∫[a to b] f(x) dx = ∫[a to b] f(a+b-x) dx

B. Applications and Limitations

These properties simplify calculations and provide consistency checks for physical and geometric results.

• Application: Symmetry can halve the calculation for even loading or temperature distributions and eliminate balanced odd contributions over symmetric intervals.

• Signed quantity: A definite integral measures net accumulation, so it is not automatically the total geometric area. Total area requires splitting at zeros and integrating (|f(x)|).

• Limitation: Symmetry rules require both the correct function parity and symmetric limits; an even or odd function integrated over a nonsymmetric interval does not receive the same simplification.