De nition of the De nite Integral - University of Illinois ...

Definition of the Definite Integral

If f is defined on [a, b], we divide [a, b] into n subintervals of equal width x = (b - a)/n. We let x0, x1, x2, . . . , xn be the endpoints of these subintervals and we let x1, x2, x3, . . . , xn be any sample points in these subintervals. Then the definite integral of f from a to b is

b

n

a

f (x) dx = lim

n

f (xk)x

k=1

provided that this limit exists and gives the same value for all possible choices of sample points. If it

does exist, we say that f is integrable on [a, b].

Theorem: If f is continuous on [a, b], or if f has only a finite number of jump discontinuities, then f is

b

integrable on [a, b]; that is, the definite integral f (x) dx exists.

a

Theorem: If f is integrable on [a, b], then

b

n

a

f (x) dx = lim

n

f (xk)x

k=1

b-a where x = n and xk = a + kx

Properties of the Definite Integral (assume f and g are continuous)

a

b

? f (x) dx = - f (x) dx

b

a

a

? f (x) dx = 0

a

b

? c dx = c(b - a)

a

b

b

? cf (x) dx = c f (x) dx

a

a

b

b

b

? [f (x) + g(x)] dx = f (x) dx + g(x) dx

a

a

a

b

b

b

? [f (x) - g(x)] dx = f (x) dx - g(x) dx

a

a

a

b

c

b

? f (x) dx = f (x) dx + f (x) dx

a

a

c

b

? If f (x) 0 for a x b, then f (x) dx 0

a

b

b

? If f (x) g(x) for a x b, then f (x) dx g(x) dx

a

a

b

? If m f (x) M for a x b, then m(b - a) f (x) dx M (b - a)

a

Fundamental Theorem of Calculus: Suppose f is continuous on [a, b].

x

? Part 1: If g(x) = f (t) dt, then g (x) = f (x).

a

b

? Part 2: f (x) dx = F (b) - F (a) where F is any antiderivative of f (that is, F = f ).

a

b

Part 2 is sometimes written as F (x) dx = F (b) - F (a) and called the Net Change Theorem since

a

the definite integral of a rate of change is equal to the net change.

List of Indefinite Integrals

? k dx = kx + C

? xn dx = xn+1 + C n+1

?

x-1 dx =

1 dx = ln |x| + C

x

? ex dx = ex + C

? ax dx = ax + C ln (a)

(where k is any constant) (where n is any constant except -1)

(where a is any positive constant except 1)

? sin (x) dx = - cos (x) + C

? cos (x) dx = sin (x) + C ? sec2 (x) dx = tan (x) + C ? csc2 (x) dx = - cot (x) + C

? sec (x) tan (x) dx = sec (x) + C

? csc (x) cot (x) dx = - csc (x) + C

1 ? 1 + x2 dx = arctan (x) + C

1

?

dx = arcsin (x) + C

1 - x2

1

?

dx = arcsec (x) + C

x x2 - 1

? sinh (x) dx = cosh (x) + C

(hyperbolic function to be discussed after third test)

? cosh (x) dx = sinh (x) + C

(hyperbolic function to be discussed after third test)

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