Process for finding a definite integral

[5ymmetrica1]

Homework Statement

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

Homework Equations

∫₀¹ (1+x+x²) dx

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?

 

[Fredrik]

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.

 

[5ymmetrica1]

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

where would I go from there if so?

 

[Fredrik]

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

 

[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?

 

[Fredrik]

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.

 

[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