Calculus II: Fundamental Theorem of Calculus
Applications

1.Changing the limits of integration

Changing the bounds of integration results from the change of the variable of integration.

If we want to solve the following definite integral

	b
∫		f(x) dx
	a

and we find that it is easier to do it by changing the variable x into h(x), then dx into h'(x) dx , the integrale becomes

	h(b)		h(b)
∫		g(h(x)) h'(x) dx = ∫		g(u) du
	h(a)		h(a)

with u = h(x).

2. Example

Let the definite integral

	1
∫		(2x + 1)2 dx .
	0

By the change
u = 2 x + 1 = h(x), so
du = 2 dx = h'(x) dx,

the integral becomes :

	3		3
∫		(1/2) u2 du = [(1/6) u3]	=  (1/6)[33 - 1] = 13/3
	1		1

3. Properties

3.1 Even function

If f is even, that is f(- x) = f (x)
in the interval [a, b], we always have:

	b		b
∫		f(x) dx = 2 ∫		f(x) dx
	a		0

3.2 Odd function

If f is odd, that is f(- x) = - f (x)
in the interval [a, b], we always have:

	b
∫		f(x) dx  =  0
	a

5. Exercises