Integration is the reverse process of differentiation. Given a derivative f'(x), integration recovers the original function f(x) (plus a constant). Integration also computes areas under curves, making it the foundation for applications in physics, engineering, economics, and probability.
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Key Concepts
Indefinite Integral: The integral of f(x) with respect to x, written as integral of f(x) dx = F(x) + C, where F'(x) = f(x) and C is the constant of integration.
- Standard Integrals:
- integral of xn dx = xn+1/(n+1) + C (n not equal to -1)
- integral of 1/x dx = ln|x| + C
- integral of ex dx = ex + C
- integral of sin x dx = -cos x + C
- integral of cos x dx = sin x + C
- integral of sec2 x dx = tan x + C
- integral of cosec2 x dx = -cot x + C
- integral of sec x tan x dx = sec x + C
- integral of 1/√(1-x2) dx = sin-1 x + C
- integral of 1/(1+x2) dx = tan-1 x + C
Methods of Integration:
1. Substitution: Replace u = g(x) so du = g'(x) dx. Transforms the integral into a simpler form.
2. Integration by Parts: integral of u · v dx = u · (integral of v dx) - integral of [u' · (integral of v dx)] dx.
Choose u using ILATE order: Inverse trig, Logarithm, Algebraic, Trigonometric, Exponential.
3. Partial Fractions: Decompose rational functions P(x)/Q(x) into simpler fractions before integrating. Used when Q(x) factors into linear or irreducible quadratic factors.
- Special Integrals:
- integral of 1/(x2 - a2) dx = (1/2a) ln|(x-a)/(x+a)| + C
- integral of 1/(a2 - x2) dx = (1/2a) ln|(a+x)/(a-x)| + C
- integral of 1/(x2 + a2) dx = (1/a) tan-1(x/a) + C
- integral of 1/√(x2 - a2) dx = ln|x + √(x2-a2)| + C
- integral of 1/√(a2 - x2) dx = sin-1(x/a) + C
- integral of √(a2 - x2) dx = (x/2) · √(a2-x2) + (a2/2) · sin-1(x/a) + C
Definite Integral: integral from a to b of f(x) dx = F(b) - F(a) (Fundamental Theorem of Calculus).
- Properties of Definite Integrals:
- integral from a to b = - integral from b to a
- integral from a to b of f(x) dx = integral from a to b of f(a+b-x) dx
- integral from 0 to 2a of f(x) dx = 2 · integral from 0 to a f(x) dx, if f(2a-x) = f(x); = 0 if f(2a-x) = -f(x).
- integral from -a to a of f(x) dx = 2 · integral from 0 to a f(x) dx if f is even; = 0 if f is odd.
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Worked Examples
Evaluate integral of (3x2 + 2x + 1) dx.
= x3 + x2 + x + C.
Evaluate integral of x · ex dx (by parts).
Let u = x, v = ex. Then u' = 1, integral of v = ex.
= x · ex - integral of 1 · ex dx = x · ex - ex + C = ex(x - 1) + C.
Evaluate integral of (2x)/(x2 + 1) dx.
Substitute u = x2 + 1, du = 2x dx.
= integral of 1/u du = ln|u| + C = ln(x2 + 1) + C.
Evaluate integral from 0 to pi/2 of sin x dx.
= [-cos x] from 0 to pi/2 = -cos(pi/2) + cos(0) = 0 + 1 = 1.
Evaluate integral of 1/(x2 - 4) dx.
= integral of 1/((x-2)(x+2)) dx. Partial fractions: 1/((x-2)(x+2)) = A/(x-2) + B/(x+2).
Solving: A = 1/4, B = -1/4.
= (1/4) ln|x-2| - (1/4) ln|x+2| + C = (1/4) ln|(x-2)/(x+2)| + C.
Evaluate integral from 0 to pi of x sin x dx (by parts).
u = x, v = sin x. integral of v = -cos x.
= [-x cos x] from 0 to pi + integral from 0 to pi of cos x dx
= (-pi · cos(pi) - 0) + [sin x] from 0 to pi
= pi + (0 - 0) = pi.
Use the property to evaluate integral from 0 to pi/2 of sin2 x dx.
Using the property integral from 0 to pi/2 f(x) dx = integral from 0 to pi/2 f(pi/2 - x) dx:
Let I = integral from 0 to pi/2 sin2 x dx. Also I = integral from 0 to pi/2 cos2 x dx.
Adding: 2I = integral from 0 to pi/2 1 dx = pi/2. So I = pi/4.
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Common mistakes
- Forgetting the constant of integration C in indefinite integrals.
- Applying limits incorrectly after substitution in definite integrals (always change limits when substituting, or revert to original variable).
- Incorrect ILATE choice in integration by parts, leading to a more complicated integral.
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Summary
Integration reverses differentiation and computes areas. The three main techniques are substitution, integration by parts, and partial fractions. Definite integrals use the Fundamental Theorem and have useful symmetry properties that simplify calculations.