What is the function h(x) in this case?

4 1 h(x) dx + 5 4 h(x) dx becomes

5 1 h(x) dx

Sry, ∫(upper limit: 5, lower limit: 1) h(x) = 4

So it’s 12+4=16?

Yes it is

There is something not right in the given expression.

Correct is

integrate (from 1 to 4)
OPEN BRACKET h(x) + 4 CLOSE BRACKET
dx

Without the open bracket and the close bracket to cover the h(x) + 4, the presentation is wrong.

The question is poorly presented by the setter.

Not true. Exam papers have always presented like this

It is when the student is writing his working that he is expected to put the brackets

So far the exam papers I come across have always put the brackets in though; it’s meant to be placed there.

In the A Levels it would be placed as well. Otherwise, if the question is viewed by an external party, the external party may feel that A Level is capable of publishing errors in a paper

They are placed by Cambridge and some school setters just to be more clear in a sense.

When you take higher level mathematics modules (H3 or university modules) or even encounter graduate level texts or research paper, you'll see the brackets not being there very often.

It is considered optional as it is understood that the integral is taken over the whole expression (it is already between the integral symbol and the dx anyway)

Another kind of expression where brackets tend to not be there :

∫ x² cos 3x dx

In this case it is understood to mean

∫ (x² cos (3x) ) dx

When
∫(UL: b, LL: a) f(x) dx + ∫(UL: c, LL: b) f(x) dx
= ∫(UL: c, LL: c) f(x) dx

Can the same be said for
∫(UL: b, LL: a) f(x) dx - ∫(UL: c, LL: b) f(x) dx

∫ (lower a, upper b) f(x) dx + ∫(lower b, upper c) f(x) dx

= F(b) - F(a) + F(c) - F(b)

= F(c) - F(a)

= ∫ (lower a , upper c) f(x) dx

(F(a) means the value of of the integral of f(x) when x = a)

So the reverse is also true.


∫(lower a, upper c) f(x) dx - ∫(lower b, upper c) f(x) dx

= ∫(lower a, upper b) f(x) dx


We could also do this :

∫(lower a, upper c) f(x) dx - ∫(lower b, upper c) f(x) dx

= ∫(lower a, upper c) f(x) dx - (- ∫(lower c, upper b) f(x) dx

= ∫(lower a, upper c) f(x) dx + ∫(lower c, upper b) f(x) dx

= ∫(lower a, upper b) f(x) dx



http://output.to/sideway/default.aspx?qno=111000019

In this website you'll see the proofs for the various properties

But for your second example,


∫ (lower a, upper b) f(x) dx - ∫(lower b, upper c) f(x) dx

= F(b) - F(a) - ( F(c) - F(b) )

= F(b) - F(a) - F(c) + F(b)

= 2F(b) - F(a) - F(c)


So the same cannot be said for subtraction as with addition

Thank you!