Basic Integration Rules

Basic Integration Rules



Multiplication by a Constant: ∫cf(u) du = c∫f(u) du

Sum Rule: ∫[f(u) + g(u)]du = ∫f(u)du + ∫g(u)du

Difference Rule: ∫[f(u) - g(u)]du = ∫f(u)du - ∫g(u)du

The definite Integral - Definition

If F(x) is the antiderivative of a function f(x) continuous on (a,b)which means F'(x) = f(x) (a is less than x is less than b), then

∫f(x)dx {from a to b} = F(x) from a to b = F(b) - F(a)

Evluation of Definite Integrals

To find ∫f(x)dx from a to b

Find indefinite integral of ∫f(x)dx = ф(x)

Evaluate ф(b) and ф(a)

Calculate ф(b) - ф(a)

Geometric interpretation of definite integral

the definite integral represents the algebraic sum of the areas of the figures bounded by

the graph of the function y = f(x)
the x axis
the straight line x =a and x = b.

if the curve goes above and below the x axis in the interval a to b, the areas of above the x axis enter this sum with a plus sign, while those below the x axis enter it with a minus sign.

Evaluation of Definite Integrals by Substitution

If a substitution is done in a definite integral, such a substitution needs to be effected at three places.

1. in the integrand

2. in the differential

3. in the limits

Properties of definite integrals

Properties of definite integrals

1. ∫_a^b f(x)dx = -∫_b^a f(x)dx



2. ∫_a^b f(x)dx = ∫_a^c f(x)dx + ∫_c^b f(x)dx


3. ∫₀^a f(x)dx = ∫₀^a f(a-x)dx

4. If f(-x) = f(x) (means f is an even function), then
∫_-a^a f(x)dx = 2∫₀^a f(x)dx

5. If f(-x) = -f(x) (means f is an odd function), then
∫_-a^a f(x)dx = 0

6. ∫₀^af(x)dx = ∫₀^af(a-x)dx and
∫_a^bf(x)dx = ∫_a^bf(a+b-x)dx

7. ∫₀^af(x)dx = ∫₀^a/2f(x)dx+∫₀^a/2f(a-x)dx

Due to the above relation

∫₀^af(x)dx = 0 if f(a-x) = -f(x)
∫₀^af(x)dx = 2∫₀^a/2f(x)dx if f(a-x) = f(x)

8. If f is continuous on [a,b], then the integral function defined by g(x) = ∫_a^xf(t)dt for x Є [a,b]is derivable on [a,b], and g'(x) = f(x) for x Є [a,b].


9. If f(x0 is periodic with period T then

∫_a^bf(x)dx = ∫_a+nT^b+nTf(x)dx, where n is an integer.

In particular

∫₀^nTf(x)dx = n∫₀^Tf(x)dx






If m and M are the smallest and greatest values of a function f(x) on an interval [a,b], then m(b-a)≤∫_a^bf(x)dx≤M(b-a)

Integral Function

If f(x) is a continuous function defined on [a,b]m then a function ф(x) defined by

ф(x) = ∫f(t)dt, (a to x); x belongs to [a,b] is called the integral function of the function f.