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
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 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 Antiderivative: F(x) = x² + x
F(3) − F(1) = (9 + 3) − (1 + 1) = 12 − 2 = 10
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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).
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
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 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 For a continuous probability distribution with density f(x):
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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.