How do you test the improper integral #int x^(-3/2) dx# from #[0, oo)# and evaluate if possible?

Answer 1

The integral is divergent

The improper integral is calculated in #2# parts :
# lim_(q->0^+)int_q^1x^(-3/2)dx+lim_(p->oo)int_1^p x^(-3/2)dx#

First part

#lim_(q->0^+)int_q^1x^(-3/2)dx=lim_(q->0^+)[x^(-3/2+1)/(-3/2+1)]_q^1#
#=lim_(q->0^+)[x^(-1/2)/(-1/2)]_q^1#
#=lim_(q->0^+)[2x^(-1/2)]_1^q#
#=lim_(q->0^+)(2q^(-1/2)-2)#
#=lim_(q->0^+)(2/sqrtq-2)#
#=+oo#

This part is divergent, therefore the integral is divergent

Second part

#lim_(p->oo)int_1^p x^(-3/2)dx=lim_(p->oo)[x^(-3/2+1)/(-3/2+1)]_1^p#
#=lim_(p->oo)[x^(-1/2)/(-1/2)]_1^p#
#=lim_(p->oo)[2x^(-1/2)]_p^1#
#=lim_(p->oo)(2-2p^(-1/2))#
#=lim_(p->oo)(2-2/sqrtp)#
#=2#

This part is convergent.

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Answer 2

To test the improper integral (\int_0^\infty x^{-3/2} , dx), you use the limit approach. First, integrate the function from 0 to a finite value (b), then take the limit as (b) approaches infinity. To evaluate:

[ \begin{aligned} \int_0^b x^{-3/2} , dx &= \left[ \frac{x^{-1/2}}{-1/2} \right]0^b \ &= \left[ -2x^{-1/2} \right]0^b \ &= -2 \left( \frac{1}{\sqrt{b}} - \lim{x \to 0} \frac{1}{\sqrt{x}} \right) \ &= -2 \left( \frac{1}{\sqrt{b}} - \lim{x \to 0^+} \frac{1}{\sqrt{x}} \right) \ &= -2 \left( \frac{1}{\sqrt{b}} - \infty \right) \ &= -2 \left( 0 - \infty \right) \ &= 2 \infty \ &= \infty \end{aligned} ]

Since the integral diverges, it does not have a finite value.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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