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.
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:
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∫ 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
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 DAQN/rzygdx1gBmhWBd9sNh9Nq3FzJ8jt2rQf0gkPLNPtGG2O7wx377sQ86GdGIHQTaXFuxI6pNAjFKZKZtJbzuAD/fNx5VIbPfP4AzK9oRFTrsJltEB/qi+q5bnwk59QkYh+FqnOEEKuWcDJ3LmivYfPS/qh7Qv/tKXhqzDU5o2CH/I2XX6Hm2DRHa3vf7vzIgg+EaEhA1ZjlSMG5VmlbhI8HTJILqUqLdVfi7r++AocezY8wtbqAKRfL2mmdiILF2lqShZKpYLZ4BQZYngfeYzTI2pXcDhx0lgDLY66owLUpz2jjNF82/4Tl3HsmwTEqg+L0wokR8uUzD+s4OPdxdz86coNH+LGDCYhGVCf2K5gpFYb6hD+AmOVx6kRCYkdzhwB4pdXlLmuGFFJV99oxGsR7wiEqVmwN3dQfOwyKEtIQ5CNuiXkD49f+oMY4VXJi5iMCUGdkNISFZFGa9IUwrZlRwrzRUKYV+/KABKJJBxIp4uKGwcZWe81+oz5o2rDJi5QfUKoN0eQKU53bS5VIV7E5alZHFZaOQtHPnTT2E9+gNBtaARsMGRIR/zbsnBglsfmoLa/6ZyW0xn2yGDZj+wde00tYaLKAsiHB6ufzkZusBOCdBQ8qnMkoaQYtlauq9SH7NIH1RgVhPjQKBoQkDPrKgjMWfwtU9EBjxMU8GkjQ90XIZveLyATNrYBI4/Ug92b2Gyl04MHXDEtSvJO/BWNLd3K0wwiha3OLtthaZjOjIqjyv2nkLzwFQLl8MFDQAxR5W9sJ1i58cSuQqErNHgxuxkXSMMNevkVRf8U3BOMBwbUpHYOgQrxzgN0GE9i9sgpQFihR2F9xJpcQfuMjcnqnQGAFTTLaQNzEnE7ovbto4Lj45pM4txVYV4TjDRXqZpoRrgdW0FaohfLUqGmr4L9lPv9 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
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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.