Integrating x^2 * [1+(sinx)^2007] from 1 to -1

[transgalactic]

x^2 * [1+(sinx)^2007]dx

the integral is from 1 to -1

i tried to solve it by parts but its too complicated

 

[quantumdude]

Have you learned anything about integrating even and odd functions?

 

[rocomath]

Have you learned anything about integrating even and odd functions?

Can you tell me a little more? Thanks :-]

 

[Mathdope]

Even functions satisfy f(-x) = f(x). Odd functions satisfy f(-x) = -f(x). Note the interval over which you are integrating makes this rather useful.

 

[transgalactic]

thats not an odd functon my function is composed of two different function
which are different from one another
f(-x) = -f(x)

cosx is an odd function ,
but here i am not sure
if i knew its odd then we need to split it into two different intervals
and still find the integral

i tried to solve it by parts but that's not working

 

[TheoMcCloskey]

thats not an odd functon

Indeed - but it is a linear combination of two functions - one even and one odd.

 

[Gib Z]

Think of the integrand as one entire function, not the composition of separate ones. Check if its even or odd. If you haven't run into integrating odd and even functions before, you can interpret it algebraically or geometrically.

Algebraically, split the integral over the intervals -1 to 0 and 0 to 1. Then use the odd function property.

Geometrically, do you see any symmetry when you graph the functions?

 

[HallsofIvy]

By symmetry, if f(x) is an even function,
$$\int_{-a}^a f(x)dx= 2\int_0^a f(x)dx$$
If f(x) is an odd function,
$$\int_{-a}^a f(x)dx= 0$$

Write your integrand as x²+ x² sin²⁰⁰⁷(x).

What is (-x)²? What is (-x)² sin²⁰⁰⁷(-x)?

 

[transgalactic]

my problem is to solve the actual function to find the previos function
on which we made the derivative

by parts is not working

 

[silver-rose]

Read HallsofIvy's post.

Expand the product out, and take the integral of each of the terms, using the idea of odd and even functions.

Why would u use by-parts in the first place? What are you trying to reduce to zero power? If you're trying to eventually take x^2 down to x^0, I don't see how you can in turn, integrate sin(x)^2007.

 

[transgalactic]

i read the post and in the end i am still required to solve
x^2 * (sinx)^2007

and i don't have any clue how to do it

 

[Gib Z]

$$f(x) = x^2 \cdot \sin^{2007} (x)$$

$$f(-x) = (-x)^2 \cdot \sin^{2007} (-x) = x^2 \cdot \left( - \sin^{2007} x\right)= - x^2 \cdot \sin^{2007} (x) = - f(x)$$

Repeating;
$$f(-x) = - f(x)$$.

$$\int^a_{-a} f(x) dx = \int^0_{-a} f(x) dx + \int^a_0 f(x) dx$$.

In the first integral, letting x = -u, dx = - du. The bounds change from 0 to 0, and -a to a.

$$\int^0_{a} - f(-u) du = \int^0_a f(x) dx = - \int_0^a f(x) dx$$.

I probably shouldn't have given you the whole answer, but I trust that you will read over the solution several times so you understand it perfectly, or else I just did a misdeed.

 

[transgalactic]

but you didnt finish it
the main problem is to solve f(x)dx
from a to 0
or the other one

iknow of the process of spliting the interval
the main problem for me is to solve this integral f(x)dx

i tried to solve it in part ,in substitution
nothing works

 

[Gib Z]

Do you just want an anti derivative or do you want the smart way?

I practically did it for you!

$$\int^a_{-a} f(x) = \int^a_0 f(x) dx - \int^a_0 f(x) dx$$.What is the right hand side equal to?

 

[transgalactic]

it equals to zero
wow
you switched the intervals

 

[Gib Z]

=] So you fine with this now? We can go over other examples if you need to.

 

[transgalactic]

thanks

 

[Defennder]

I'm wondering, does anyone know how to evaluate the function as an indefinite integral? Any hints?

 

[Gib Z]

The anti derivative will be ugly, if it exists in elementary form.

 

[transgalactic]

why its zero?

By symmetry, if f(x) is an even function,
$$\int_{-a}^a f(x)dx= 2\int_0^a f(x)dx$$
If f(x) is an odd function,
$$\int_{-a}^a f(x)dx= 0$$

Write your integrand as x²+ x² sin²⁰⁰⁷(x).

What is (-x)²? What is (-x)² sin²⁰⁰⁷(-x)?

because if we draw the function we get two areas which are
opposite one to another

bu still these areas exists and we need to count them
they both equal to each other
so we need to find the area of one of them
and multiply the result by 2

?

why its zero
zero is a mistake
we need to find the area of one of them
and multiply the result by 2

 

[HallsofIvy]

i read the post and in the end i am still required to solve
x^2 * (sinx)^2007

and i don't have any clue how to do it

but you didnt finish it
the main problem is to solve f(x)dx
from a to 0
or the other one

iknow of the process of spliting the interval
the main problem for me is to solve this integral f(x)dx

i tried to solve it in part ,in substitution
nothing works

bu still these areas exists and we need to count them
they both equal to each other
so we need to find the area of one of them
and multiply the result by 2

?

why its zero
zero is a mistake
we need to find the area of one of them
and multiply the result by 2

That's NOT what you originally said! Your original post said you were to find the definite integral from 1 to -1, not the anti-derivative (indefinite integral). And, because your function is odd, that's easy.

 

[fermio]

put inside 3 ( x^2 (1+Sin[ x] )^5 ; x^2 (1+Sin[ x] )^10 ; x^2 (1+Sin[ x] )^20 ) values into http://integrals.wolfram.com/index.jsp and you will see that it's imposible even for supercomputer integrate x^2 (1+Sin[ x] )^2007.

 

[transgalactic]

i asked about this and i was told that
i am confusing two cases
that in this case we do strate integral

thats why we get 0

but there is another type of question for which we need to look into the graph
of the given function and make a sum of each area

can you make an example for a question that we need to solve
it by the second way

 

[transgalactic]

the definition of odd

Indeed - but it is a linear combination of two functions - one even and one odd.

as you sayd this is a linear combination of both odd and even

so i can guess that the hole function is odd

what if i have a function with 2 odds and one even

whay now??
is it odd or even??

what is the rule in mixed functions(when there is no f(-x)=-f(x) f(-x)=f(x))??