Question

DIVERGES OR COVERGES?

Answer 1

\[\int\limits_{1}^{\infty}(\frac{ 1 }{3\sqrt{x} }-\frac{ 1 }{ \sqrt{9x+7}})dx\]

Answer 2

if we integrate without putting limits, we get
-2/9 (sqrt(9x +7) - 3 sqrt(x))

now on putting limits,

=> limx->inf [ -2/9 (sqrt(9x +7) - 3 sqrt(x)) ] + 2/9

which just simplifies to 2/9 as the final asn

Answer 3

so convergant i'd say.. please correct me if i went wrong anywhere..

Answer 4

integrating this, I get [2/3 * x^(1/2) - 2/9 * (9x+7)^(1/2)] which needs to be evaluated at infinity and 1, to get the value of the definite integral (or whatever it's called). It comes out as infinity I think.

Answer 5

actually... (I'm tired...) maybe not infinity...

Answer 6

I guess
This argument won't work here...
=> limx->inf [ -2/9 (sqrt(9x +7) - 3 sqrt(x)) ] + 2/9 is finite so it is finite because area under a curve of 1/x is infinite from x=1 to infinity though 1/x is finite as x tends to infinity.

Answer 7

What is [ -2/9 (sqrt(9x +7) - 3 sqrt(x)) ] + 2/9 from? Why is there the constant 2/9? This looks to me like an error.

Answer 8

I get 2/3 * x^(1/2)-(2/3) - 2/9 * (9x+7)^(1/2)+(8/9)

Answer 9

how to evaluate with infinity , the answer will be (infinity - infinity)

Answer 10

I guess we have to bound this using by upper Riemann sums...