Question

evaluate the integral to determine convergence or divergence

Answer 1

\[\int\limits_{1}^{\infty}\frac{ x^9 }{ x^{10}+4 }\]

Answer 2

try \(u = x^{10}+4\)

Answer 3

what about the top?

Answer 4

well what would du be?

Answer 5

10x^9 ?

Answer 6

\[u=x^{10}+4 \rightarrow \frac{du}{dx}= 10x^{9}\]

Answer 7

\[du=10x^{9}dx\]

Answer 8

so you basically have the du you just need a 10, right?

Answer 9

is it 1/10?

Answer 10

\[\frac{ 1 }{ 10 } \int\limits_{1}^{\infty}\frac{ 1 }{ u }du\]

Answer 11

well, we multiply by one 1=10/10 pull the 10 inside the integral to obtain the du and leae the 1/10 outside

Answer 12

yes

Answer 13

so, i take the anti derivative of u now?

Answer 14

yes

Answer 15

\[\frac{ 1 }{ 10 }[\frac{ 1 }{ 2u^2 }] ,1--> \infty\]?

Answer 16

is it that the 1/10 makes this divergent?

Answer 17

well notice that the denominator is bigger than the numerator, so it will, therefore, grow faster on the bottom.

Answer 18

yes, what does that mean with the integral test?

Answer 19

does it mean that since the answer is technically "infinity" that means "divergent"?

Answer 20

I would say it is divergent

Answer 21

you can test it via ratio test

Answer 22

but we know that 1/x --> 0 when x--->inf
Just apply the same principle

Answer 23

idk what the ratio test is, ill have to look that up

Answer 24

i was just trying to determine if \[\frac{ 1 }{ 10 }(\infty)=\] divergent in a general sense

Answer 25

if you want to determine whether the integral is convergent or divergent, you have to evaluate first and see what you get.
What did you get when you evaluated the integral?

Answer 26

1/10 * infinity= infinty

Answer 27

\[\int \frac{1}{u}du = \ln |u| + C\]

right ?

Answer 28

oh yea..