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

The integral is divergent

First part

This part is divergent, therefore the integral is divergent

Second part

This part is convergent.

By signing up, you agree to our Terms of Service and Privacy Policy

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.

By signing up, you agree to our Terms of Service and Privacy Policy

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.

- Determine whether the series # sum_(n=1)^oo (2n^2 +3n)/sqrt(5+n^5)# is convergent or divergent. How do i tell which comparison test to use?
- How do you determine the convergence or divergence of #Sigma ((-1)^n n!)/(1*3*5***(2n-1)# from #[1,oo)#?
- How do you use the ratio test to test the convergence of the series #∑ 3^n/(4n³+5)# from n=1 to infinity?
- How do you test the improper integral #int x^(-1/3) dx# from #(-oo, oo)# and evaluate if possible?
- How do you determine if #a_n=(1+n)^(1/n)# converge and find the limits when they exist?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7