Integration stuff

Question

Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as "DNE".

∫ oo on top and 2 on bottom (9)/(x+2)^3/2 dx

Answer 1

Hi,

∫ oo on top and 2 on bottom (9)/(x+2)^3/2 dx

your question is :

9∫(x+2)^(-3/2) dx as you know ∫u^n du = 1/(n+1) u^(n+1)

now u = x+2 n = -3/2 dx = du so the answer of integral will be (-2)(x+2) ^ -1/2

so 9 ( -2) / √x+2 from 2 to ∞ ====> so with replacing ∞ instead of x you have zero then replace 2 instead of x

you will have 9

so the answer of that integral is 9 and convergent.

I hope it is useful,

Minoo

Answer 2

∫_{(2 to ∞)} [9 / (x+2)^3/2]dx = 9lim_b→∞ ∫_{(2 to b)} (x+2)^-3/2dx = 9lim_b→∞ [-2/√(x+2)]_{(2 to b)}

= -18lim_b→∞ [1 / √(b + 2) - 1 / 2] = -18(0 - 1/2) = 9

The integral converges to 9.