In This Lesson 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 Antiderivatives The Definite Integral 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 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
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.
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
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 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 u = x, dv = eˣ dx → du = dx, v = eˣ
= xeˣ − ∫ eˣ dx = xeˣ − eˣ + C
3. Partial Fractions Decompose a rational function into simpler fractions. Requires factoring the denominator first:
∫ 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 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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 These connect integration to 3D geometric shapes — computing volumes that geometry formulas alone can't handle.
Arc Length L = ∫ₐᵇ √(1 + [f'(x)]²) dx
Probability For a continuous probability distribution with density f(x):
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 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.