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):
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∫ 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 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 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 :
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Example: ∫ x·eˣ dx 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 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 Area = ∫ₐᵇ [f(x) − g(x)] dx where f(x) ≥ g(x) on [a,b]
Volume of Revolution 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
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.
Arc Length 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 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.