- #1

- 479

- 20

$$\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$$

What exactly are the prerequisites for this property?

Should ##c## be a member of ##[a,b]##? Should the function ##f## be defined at ##c##?

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- Thread starter PFuser1232
- Start date

- #1

- 479

- 20

$$\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$$

What exactly are the prerequisites for this property?

Should ##c## be a member of ##[a,b]##? Should the function ##f## be defined at ##c##?

- #2

- 210

- 12

c need not be within [a,b]. Itis because c is going to be subtracted

- #3

- 479

- 20

Could you please elaborate? And what about the existence of ##f(c)##?c need not be within [a,b]. Itis because c is going to be subtracted

- #4

- 210

- 12

∫ f'(x) = f(x) + c

And within limit b to a it will be f(b)-f(a)

Look at the right hand side, there is a sum of two integrals having limit c to a, and b to c.

So R.H.S={f(c)-f(a)} + {f(b)-f(c)}

So the f(c) is subtracted and finally it is f(b)-f(a) like the L.H.S

So, c need not be within [a,b]

I am not sure if f(c) should be defined.

As f(c) is getting subtracted so it does not matter. But generally an equation is used for solving problems. If you take such value of c where the function is not defined, you cannot use the equation to solve the problem.

- #5

HallsofIvy

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