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Integration is the reverse of differentiation. The integral of a function f(x) finds the antiderivative F(x) such that F'(x) = f(x). Integration also computes the area under a curve between two bounds — one of the most useful applications in science and engineering.
An indefinite integral ∫f(x)dx = F(x) + C includes a constant C because infinitely many antiderivatives differ only by a constant. A definite integral ∫[a→b] f(x)dx = F(b) − F(a) gives the exact signed area between the curve and the x-axis from x = a to x = b.
Key integration rules: the power rule (∫xⁿdx = xⁿ⁺¹/(n+1) + C for n ≠ −1), ∫eˣdx = eˣ + C, ∫sin(x)dx = −cos(x) + C, ∫cos(x)dx = sin(x) + C, and ∫(1/x)dx = ln|x| + C.
Worked example: ∫(4x³ − 6x + 5)dx = x⁴ − 3x² + 5x + C. For the definite integral ∫[0→2](4x³ − 6x + 5)dx = [x⁴ − 3x² + 5x]₀² = (16 − 12 + 10) − 0 = 14.
∫[a→b] f(x)dx = F(b) − F(a), where F'(x) = f(x)
Common Integrals
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