Question

I need a starting point with the integral below...

Answer 1

\[\int\limits_{-1}^{1} x^8 \sin(x) dx\]

just a starting tip would be nice. I want to try and solve it on my own

Answer 2

start with integration by parts

Answer 3

This looks like you would want to use integration by parts. Since you have \(x^8\) I would suggest using the table method.

Answer 4

got alot of tabling there :)

but then sin just jostles back and forth so its not too complicated

Answer 5

oh lord what a pain this will be
parts 8 times?
amistre, if you are lucky, will do the table method ala escalante

Answer 6

Well i tried:

\[u = sinx\]
\[du = cosx\]
\[dv = x^8\]
\[v = (1/9) (x^9)\]

so if i integrate by parts:

\[1/9(x^9)(sinx) - \int\limits_{-1}^{1} 1/9 (x^8) (cosx) dx\]

and now i'm back to where i started

Answer 7

\[\sum_{n=8}^{0} \frac{d^n}{dx^n}x^8\int^n sin(x)\]
thats terrible notation

Answer 8

and if i try the other way, i'll end up with eight different integrals and all of then will have to be evaluated from -1 to 1. I'm wondering if there is a shorter method that i'm not thinking of.

Answer 9

the function is odd

Answer 10

sin(x)

+ x^8 -cos(x)
- 8x^7 -sin(x)
+8.7x^6 cos(x)
-8.7.6x^5 sin(x)
+8.7.6.5x^4 -cos(x)
-8.7.6.5.4x^3 -sin(x)
+8.7.6.5.4.3x^2 cos(x)
-8.7.6.5.4.3.2x sin(x)
+8.7.6.5.4.3.2 -cos(x)

-0

product those and simplify :)

Answer 11

the answer is therefore 0

Answer 12

thank you everyone! don't you just love calculus!

Answer 13

I think Zarkon is right. An odd function integrated from -a to a will always be 0. Since sine is an odd function, and \(x^8\) is even the entire thing is odd.

Answer 14

i dont think thats how you determine an even odd function

Answer 15

x^8 sin(x) = (-x)^8 sin(-x)

x^8 sin(x) = - x^8 sin(x)

therefore odd

but i could be mistaken

Answer 16

i never remember shorcuts too well :)

Answer 17

(even times even)=even
(odd times odd)=even
(even times odd)=odd
(odd times even)=odd
So if you know what both functions are, it's straightforward. Otherwise, you would need to check it algebraically.

Answer 18

yea i was thinking about it too:

(-x)^8 = x^8

sin(-x) = -sin(x)

since (x^8)(sin(-x)) = -(x^8)(sin(x)) I argued the function must be odd.

Answer 19

yep

Answer 20

i graphed it with Wolfram Alpha too just to be sure.