Given that definite integral of f(x) from x = 1 to x = 2 is 2 and definite integral of f(x) from x = 1 to x = 4 is - 1find integral of f(x) from x...

If f(x) is defined and continuous over the interval [a,
b], except maybe at a finite number of points, we'll write Int f(x)dx from a to b
as:

Int f(x)dx (a->b)  = Int f(x)dx (a->c) +
Int f(x)dx(c->b)

We'll put the endpoints of the
interval [a,b] as:

a = 1 and b =
4.

Int f(x)dx (1->4)  = Int f(x)dx (1->2) +
Int f(x)dx(2->4)

We'll subtract both sides by Int
f(x)dx (1->2) and we'll use the symmetric
property:

Int f(x)dx(2->4) = Int f(x)dx
(1->4) - Int f(x)dx
(1->2)

We'll substitute Int f(x)dx (1->4) =
-1 and

Int f(x)dx (1->2) = 2 and we'll
get:

Int f(x)dx(2->4) = -1 - 2 =
-3

So, the definite integral of the function
f(x), from x = 2 to x = 4
is:

Int f(x)dx(2->4) =
-1 - 2 = -3