Begin2.DVI

Definite integrals

General integration properties

1. IfdFdx(x)=f(x), then

∫b

a

f(x)dx=F(x)|ba=F(b)−F(a)

2. ∫∞

0

f(x)dx= limb→∞

∫b

0

f(x)dx,

∫∞

−∞

f(x)dx= limb→∞

a→−∞

∫b

a

f(x)dx

3. Iff(x)has a singular point atx=b, then

∫b

a

f(x)dx= lim→ 0

∫b−

a

f(x)dx

4. Iff(x)has a singular point atx=a, then

∫b

a

f(x)dx= lim→ 0

∫b

a+

f(x)dx

5. Iff(x)has a singular point atx=c,a < c < b, then

∫b

a

f(x)dx=

∫c−

a

f(x)dx+

∫b

c+

f(x)dx

6.

∫b

a

cf(x)dx=c

∫b

a

f(x)dx, cconstant

∫a

a

f(x)dx=0,

∫b

0

f(x)dx=

∫b

0

f(b−x)dx

∫b

a

f(x)dx=−

∫a

b

f(x)dx,

∫b

a

f(x)dx=

∫c

a

f(x)dx+

∫b

c

f(x)dx

7. Mean value theorems
   ∫b

a

f(x)dx=f(c)(b−a), a≤c≤b

∫b

a

f(x)g(x)dx=f(c)

∫b

a

g(x)dx, g(x)≥ 0 , a≤c≤b

∫b

a

f(x)g(x)dx=f(a)

∫ξ

a

g(x)dx

∫b

a

f(x)g(x)dx=f(b)

∫b

η

g(x)dx

a < ξ < b a < η < b

The last mean value theorem requires thatf(x)be monotone increasing and nonnegative

throughout the interval(a, b)

8. Numerical integration
   Divide the interval(a, b)intonequal parts by defining a step sizeh=b−na.

Two numerical integration schemes are

(a) Trapezoidal rule with global error−(b− 12 a)h^2 f′′(ξ)fora < ξ < b.

∫b

a

f(x)dx=h 2 [f(x 0 ) + 2f(x 1 ) + 2f(x 2 ) +··· 2 f(xn− 1 ) +f(xn)]

(b) Simpson’s 1/3 rule with global error−(b 90 −a)h^4 f(iv)(ξ)fora < ξ < b.

∫b

a

f(x)dx=^23 h[f(x 0 ) + 4f(x 1 ) + 2f(x 2 ) + 4f(x 3 ) + 2f(x 4 ) +···+ 2f(xn− 2 ) + 4f(xn− 1 ) +f(xn)]

Appendix C