Question

Determine whether integral is convergent or divergent. if convergent, evaluate.?
also, tell me which series you used


1) integral from 0 to infinity 2 / (x+2)^3 dx






2) integral from 0 to infinity 5/ (x+5)^1/5 dx

Answer 1

just let x+2= something in 1)
and x+5= something in 2)
rest should be easier ..

Answer 2

first one converges because the denominator is a polynomial of degree 3 and the numerator is a polynomial of degree 0 (it is a constant)
and \(3-0>1\) which is all you need

Answer 3

second one does not by the same reasoning \(\frac{1}{5}-0<1\)

Answer 4

But how would I test that? the answer is 1/4, but i need step by step guidance to get that. Thanks

Answer 5

first find the anti derivative, and then take the limit as \(x\to \infty\)

Answer 6

and evaluate it from 0 to infinity? It's that simple?

Answer 7

\[\int_0^{\infty}\frac{2dx}{(x+2)^3}\]
\[=\lim_{t\to \infty}\int_0^t\frac{2dx}{(x+2)^3}\]
\[=\lim_{t\to \infty}-\frac{1}{(t+2)^2}-\frac{-1}{(0+2)^2}\]
\[=0+\frac{1}{4}\]

Answer 8

thanks

Answer 9

yw