Process for finding a definite integral

1. Feb 28, 2013

5ymmetrica1

1. The problem statement, all variables and given/known data

given that 01 xndx = 1/ n+1
for integers n $\geq$ 0
calculate these integrals

2. Relevant equations
01 (1+x+x2) dx

3. The attempt at a solution
I have found the definite integral of 01 (1+x+x2) dx
to = 1.8333.... (ie. 11/6)
but all I did was use my TI-84 to find the definite integral. is it possible for me to find this solution by hand without a calculator?

2. Feb 28, 2013

Fredrik

Staff Emeritus
Yes, it is. You need to find a way to rewrite $\int_0^1(1+x+x^2)\,\mathrm{d}x$ so that you can use the formula you were given.

3. Feb 28, 2013

5ymmetrica1

could I use this?
011dx + ∫01xdx + ∫01 x2dx

where would I go from there if so?

4. Feb 28, 2013

Fredrik

Staff Emeritus
Yes, that's the correct first step. Now you just use the formula you were given on each of the three terms.

5. Feb 28, 2013

5ymmetrica1

I think I have it now thanks Fredrik!
I could use 1/1 + 1/2 + 1/3 (as 1 = x0)
Which would be 1 + 0.5 + 0.33 = 1.833

is this the correct process for working out these types of problems, just so I know in future?

6. Feb 28, 2013

Fredrik

Staff Emeritus
Yes, it is. Your book should contain a theorem that says that for all integrable functions f,g, and all real numbers a,b, we have
$$\int_a^b \left(f(x)+g(x)\right)\,\mathrm{d}x =\int_a^b f(x)\,\mathrm{d}x +\int_a^b g(x)\,\mathrm{d}x.$$ This allows you to deal with one term at a time.

7. Mar 1, 2013

5ymmetrica1

yes I see it now, along with a few similar theorems, it was located in a previous chapter of my textbook and I hadn't read it yet.
Thanks again for pointing that out, and for your help.

-5ym