Integrals

The art of accumulation — find areas, volumes, and totals by summing infinitely many infinitesimal pieces.

Antiderivatives

An antiderivative (or indefinite integral) of f(x) is any function F(x) whose derivative is f(x):

vValPg2LZ2TMYMIvhBr6k8Z6gjG3E5FSiH+6jtEiIi3aQDss0QoFKSbn3m9MMqOPNh+JsjwjvaLlS/HyBJsXlow6NRJokR2RBfG1D0igQg8fZ5OK9EMxN/iJwoB9I7dRkyhlsfGSXoIXZlcwEHCjLh9LV3+VUjCmBNVsPg5cT46QhzyUXKZKn+ph1VTGs2luCkqvl6Q4qU/g9xz6+G3IKcJWk7L/Z4UdgHTShBUqKZIOuYwb9fml5mNpwyHgYwomhNXIbRVq82wy6dK0BfuHj0nH4HB1zdGDFFjYuJrLbb6eWx+lS8TgoarRsIVGtO0ucc8IzPRfqV8o4MNgG6eEP76AH5PbNU69Qtj096UW2v3D9jZ2QbC+Vfsake7n7NKHdOHQENawKDrtUf+8y0fWKIO1hx6ficvE42dWRp2oNT+LFpQ6PJfn63owKnnUrpBGM8qu3z3n9AgKhn+j18COOj0qx56IkivL+MpWEnWi14gn39+11+FwIlKH2dKG9bR+RdDwI1360+f0RbwBCNglSDgoW6lP/p3yy4IIxHHw2DcTQyyNdU8EQ0rWqZfhAvQpjXDIiycfAvokrGCftVNkGlRPtI1XlQc73pTW4mezNt9fTsMzh7UrXVOYctFgxcLVExRk4K+JzwsWezm1kvHoIWk6miwwpGMlKY65SV/vh5nYfk+6JAFa6aHsvGCSP0jsJrVShw1P7eL80MK/DRY2iwLaOsjIbU6ni58FVtjlFBz+NuNTZgQACh5SEIxkCcEDixeW6zORIn/t3AIClced9rfJMQm87Mq6rb9gamhJokJduF/oISKFD28tYVKUQ23y2vCSWf3sipqtswcNHssixwc8LLKSBPRYlNg6/MpObq1Aiod4cT/t3sPu+6xL7QJG/42Wmx5hqCeR/fffAKVDouYhf8
∫ 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:

QxnxJFLnjl+UHPl/7BPlb/9CfAE+5LsNn6KmMg4AZdUipQZTQggG1fo22HPcPQO1TyKol77b4o7EAMsBigSr9oyfulHwLphov4Yu11nagl4wGxfTXp9qjalAb1nXSCoRjOtOmMNCaqIQ0fvOBkZURlcc77BVFHoF4UXteNWdm7z1JvC2x7ubM90K4TM0f77yFHt5yxqpxYxllOsy5yAUa5PFM9phjL0GIQ7u+qDUOKF79Z/kJl1qRPJEdSb4AYcX6lZFQ6GWSBOfDde6s81S7vIt3lVbVZX1wkQhAlWiYH+yrUO2Y7DaC/KisAHePpFiis6lCmY2MTWWAay1GFhSO/c2HpgWGk6ImsL9hLSdhPQsX+YewgWiYbM4JBT5FYdwiNXxUw1oREUTVg7OtJBAqnOcJwFmXebPjK/t/2U1LlolEd6YGhTAwY19Aqb8DJHe+YQw+uqxksBpdQsa7nqqo1PRUTazOAGqru14lZYvedRXXZFEk/9WPZtG6WEde0LmadNBT/crmB62Z0Zvl4a/2MelMEY2i9YJwrHrLUWnLnmcj7ln3yFlvYoPDM5SHayCSDL5QEYOdQthpe9a73qs4U87FiSLJho4QclXbbmlkyjyG2bvjU9e7NVRMCDCpGVNTPtqN32gOZqTKtVYgavSieIKK7rmcOUfjfPQ+bWsXtDgft9Vnio3hkF0irAGi+rde83fOokNurmbeCMkMcoPgPxBydMyOFqp6idn/rCrUzAZ1c+7dftaVuivAV0GdVfCa02k1EkamAwqZid1HBUJ7Y2CUiM9l0P8iPhMrMWDixPyNM8rZAsvEspYx+T1jEhN4HY8Kms7tttWXSC078/XXcwMkOdAF5ORsXFNsCFC5zsKZDm5HYMuUlZjSCPJbc1+oxOyUSCp50unNct1gfrkoTJla0
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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

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:

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
∫ u dv = uv − ∫ v du

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

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

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.
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