In This Lesson Antiderivatives The Definite Integral The Fundamental Theorem of Calculus Integration Techniques Applications of Integration Antiderivatives An antiderivative (or indefinite integral) of f(x) is any function F(x) whose derivative is f(x):
∫ f(x) dx = F(x) + C (where F'(x) = f(x))
The "+C" is crucial — there are infinitely many antiderivatives, differing by a constant. Some key antiderivatives:
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)
∫ 1/x dx = ln|x| + C
∫ eˣ dx = eˣ + C
∫ sin x dx = −cos x + C
∫ cos x dx = sin x + C
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The Definite Integral The definite integral computes the signed area between f(x) and the x-axis from a to b:
∫ₐᵇ f(x) dx = lim (n→∞) Σ f(xᵢ)Δx
This is a limit of Riemann sums — rectangles approximating the area. The connection to probability is direct: the probability of a continuous random variable falling in [a,b] is exactly ∫ₐᵇ f(x) dx where f is the probability density function.
The Fundamental Theorem of Calculus The FTC links differentiation and integration — two seemingly opposite operations are inverses:
Part 1: d/dx [∫ₐˣ f(t) dt] = f(x) Part 2: ∫ₐᵇ f(x) dx = F(b) − F(a) where F'(x) = f(x)
Example: Evaluate ∫₁³ (2x + 1) dx Antiderivative: F(x) = x² + x
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Integration Techniques 1. u-Substitution The integral version of the chain rule . Let u = g(x), du = g'(x)dx:
Example: ∫ 2x·cos(x²) dx 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 Let u = x², du = 2x dx
∫ cos(u) du = sin(u) + C = sin(x²) + C
2. Integration by Parts The integral version of the product rule :
∫ u dv = uv − ∫ v du
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= xeˣ − ∫ eˣ dx = xeˣ − eˣ + C
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∫ dx/(x² − 1) = ∫ [1/(2(x−1)) − 1/(2(x+1))] dx = ½ ln|x−1| − ½ ln|x+1| + C
4. Trigonometric Substitution Use trig identities to handle expressions involving √(a² − x²), √(a² + x²), or √(x² − a²).
Applications of Integration Area Between Curves 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 Area = ∫ₐᵇ [f(x) − g(x)] dx where f(x) ≥ g(x) on [a,b]
Volume of Revolution
Disk method: V = π ∫ₐᵇ [f(x)]² dx
Shell method: V = 2π ∫ₐᵇ x·f(x) dx
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Arc Length L = ∫ₐᵇ √(1 + [f'(x)]²) dx
Probability For a continuous probability distribution with density f(x):
P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx and ∫₋∞^∞ f(x) dx = 1
The normal distribution , exponential distribution, and every other continuous distribution is defined through integrals.
Integration reaches into every field: physics (
work, energy, fluid pressure ), engineering (signal processing via
Fourier transforms ), economics (consumer/producer surplus), and
linear algebra (inner products on function spaces). The concept of area under a curve is one of the most widely applied ideas in all of science.