WeBWorK 5 - Topics 10 - 12: Problem 7
(1 point)
Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as "divergent".
\[
\int_{4}^{\infty} \frac{5}{(x+3)^{3 / 2}} d x=\square
\]
Answer
The integral \(\int_{4}^{\infty} \frac{5}{(x+3)^{3 / 2}} dx\) is convergent and its value is approximately \(\boxed{3.78}\).
Step 1 ：The integral is an improper integral because it has an infinite limit. To determine whether it is convergent or divergent, we can use the comparison test or limit comparison test. However, in this case, it is easier to directly compute the integral and see if it results in a finite number or not. If the integral results in a finite number, it is convergent. If not, it is divergent.
Step 2 ：The integral is of the form \(\int_{a}^{\infty} f(x) dx\), which can be rewritten as \(\lim_{b \to \infty} \int_{a}^{b} f(x) dx\).
Step 3 ：We can compute the integral \(\int_{4}^{b} \frac{5}{(x+3)^{3 / 2}} dx\) and then take the limit as \(b \to \infty\).
Step 4 ：The integral results in a finite number, which means the integral is convergent.
Step 5 ：The integral \(\int_{4}^{\infty} \frac{5}{(x+3)^{3 / 2}} dx\) is convergent and its value is approximately \(\boxed{3.78}\).
