In This Lesson 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 Antiderivatives The Definite Integral The Fundamental Theorem of Calculus 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 Integration Techniques 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 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
More antiderivatives can be found on the formula sheet .
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.
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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
F(3) − F(1) = (9 + 3) − (1 + 1) = 12 − 2 = 10
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 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
Example: ∫ x·eˣ dx u = x, dv = eˣ dx → du = dx, v = eˣ
= 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).
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Volume of Revolution
Disk method: V = π ∫ₐᵇ [f(x)]² dx
Shell method: V = 2π ∫ₐᵇ x·f(x) dx
These connect integration to 3D geometric shapes — computing volumes that geometry formulas alone can't handle.
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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.