Calculus — The Definition and Applications of the Integral

The Antiderivative

The definition of an antiderivative:

If F’(x) = f(x), then F(x) is called an antiderivative of f(x)
Integral notation: ∫f(x)dx = F(x) + C (C: the constant of integration)

Basic integration formulas:

∫x^n dx = x^(n+1)/(n+1) + C (n ≠ −1)
∫(1/x) dx = ln|x| + C
∫e^x dx = e^x + C
∫a^x dx = a^x/ln(a) + C

Trigonometric integrals:

∫sin(x) dx = −cos(x) + C
∫cos(x) dx = sin(x) + C
∫sec²(x) dx = tan(x) + C
∫csc²(x) dx = −cot(x) + C

Linearity of integration:

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

Integration Technique 1: U-Substitution

U-substitution:

Let u = g(x), so du = g’(x)dx
Used for integrating complex composite functions

Procedure:

Set u = g(x) (the inner function)
Compute du = g’(x)dx
Rewrite the expression in x in terms of u
Integrate with respect to u
Substitute back to x

Examples:

∫2x·sin(x²)dx u = x², du = 2x dx = ∫sin(u)du = −cos(u) + C = −cos(x²) + C
∫x/(1+x²) dx u = 1+x², du = 2x dx, x dx = du/2 = (1/2)∫(1/u)du = (1/2)ln|u| + C = (1/2)ln(1+x²) + C

Integration Technique 2: Integration by Parts

Integration by parts:

Formula: ∫u dv = uv − ∫v du
Used when integrating a product of two functions

The LIATE order (for choosing u):

L: Logarithmic
I: Inverse trigonometric
A: Algebraic (x^n)
T: Trigonometric
E: Exponential
Whichever comes earlier in the list is u, the rest is dv

Examples:

∫x·e^x dx u = x, dv = e^x dx du = dx, v = e^x = xe^x − ∫e^x dx = xe^x − e^x + C
∫ln(x) dx u = ln(x), dv = dx du = (1/x)dx, v = x = x·ln(x) − ∫x·(1/x)dx = x·ln(x) − x + C

The Definite Integral

The definition of the definite integral (a Riemann sum):

∫[a to b] f(x)dx = lim(n→∞) Σf(x_i)·Δx
Intuitively: the (signed) area under the curve

Properties of the definite integral:

∫[a to b] f(x)dx = −∫[b to a] f(x)dx
∫[a to a] f(x)dx = 0
∫[a to b] f(x)dx + ∫[b to c] f(x)dx = ∫[a to c] f(x)dx
∫[a to b] [f+g]dx = ∫[a to b]f dx + ∫[a to b]g dx

Odd and even function properties (on the interval [−a, a]):

Even function f(−x) = f(x): ∫[−a to a] = 2∫[0 to a]
Odd function f(−x) = −f(x): ∫[−a to a] = 0

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus, Part 1:

If F(x) = ∫[a to x] f(t)dt, then F’(x) = f(x)
Integration and differentiation are inverse operations

The Fundamental Theorem of Calculus, Part 2:

∫[a to b] f(x)dx = F(b) − F(a)
Use any F(x) with F’(x) = f(x) to compute the definite integral

Worked examples:

∫[0 to π] sin(x)dx = [−cos(x)]₀^π = −cos(π) − (−cos(0)) = 1 + 1 = 2
∫[1 to e] (1/x)dx = [ln|x|]₁^e = ln(e) − ln(1) = 1 − 0 = 1

The mean value theorem for integrals:

There exists a c in [a,b] such that ∫[a to b] f(x)dx = f(c)(b−a)
f(c) is the average value of the integral

Applications of Integration

Area:

Area above the x-axis: A = ∫[a to b] f(x)dx (f(x) ≥ 0)
Area between two curves: A = ∫[a to b] [f(x)−g(x)]dx (f≥g)

Example: the area between y=x² and y=x

Intersection points: x²=x → x=0, x=1
A = ∫[0 to 1] (x−x²)dx = [x²/2 − x³/3]₀¹ = 1/2 − 1/3 = 1/6

Volume of a solid of revolution:

Rotation about the x-axis: V = π·∫[a to b] [f(x)]²dx (the disk method)
Rotation about the y-axis: V = 2π·∫[a to b] x·f(x)dx (the shell method)

Arc length:

L = ∫[a to b] √(1 + [f’(x)]²) dx

Average value:

The average of a function: f_avg = (1/(b−a)) ∫[a to b] f(x)dx

Frequently Asked Questions

Q. When should I use substitution versus integration by parts? A. Substitution works well when a composite function f(g(x)) appears alongside its derivative g’(x) — in other words, an integral of the form ∫f(g(x))·g’(x)dx. Integration by parts is used when two different types of functions are multiplied together — algebraic × trigonometric as in ∫x·sin(x)dx, algebraic × exponential as in ∫x·e^x dx, or a logarithm as in ∫ln(x)dx.

Q. Can a definite integral come out negative? A. Yes, it can. A definite integral is negative when the function lies below the x-axis. For example, ∫[π to 2π] sin(x)dx = −2. To find the actual “area,” you’d need to integrate |f(x)| or split the region above and below the x-axis. Remember that a definite integral is a “signed area” — it’s a directional quantity.