Math  /  Calculus

Question(1 point)
Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as "divergent". 23(x+7)3/2dx=\int_{2}^{\infty} \frac{3}{(x+7)^{3 / 2}} d x= \square Preview My Answers Submit Answers

Studdy Solution

STEP 1

What is this asking? We need to figure out if the given integral converges to a value or diverges (goes to infinity).
If it converges, we need to find that value. Watch out! Remember to carefully handle the bounds of integration, especially the infinity part.
Also, don't forget about the power rule for integration!

STEP 2

1. Rewrite the Improper Integral
2. Find the Indefinite Integral
3. Evaluate the Definite Integral

STEP 3

Let's **rewrite** this improper integral with a limit: limb2b3(x+7)3/2dx\lim_{b \to \infty} \int_{2}^{b} \frac{3}{(x+7)^{3/2}} dx This sets us up to deal with the infinity later.

STEP 4

Let's **focus** on the indefinite integral first: 3(x+7)3/2dx\int \frac{3}{(x+7)^{3/2}} dx

STEP 5

We can **rewrite** the integral as: 3(x+7)3/2dx\int 3(x+7)^{-3/2} dx This makes it easier to apply the power rule.

STEP 6

Using the **power rule** for integration (add one to the exponent and divide by the new exponent), we get: 3(x+7)3/2+13/2+1+C\frac{3(x+7)^{-3/2 + 1}}{-3/2 + 1} + C Where CC is the constant of integration.

STEP 7

**Simplifying** the exponent and the denominator: 3(x+7)1/21/2+C\frac{3(x+7)^{-1/2}}{-1/2} + C

STEP 8

**Further simplifying**: 6(x+7)1/2+C-6(x+7)^{-1/2} + C So, the indefinite integral is 6(x+7)1/2+C-6(x+7)^{-1/2} + C.

STEP 9

Now, let's **evaluate** the definite integral using the result we just found: limb[6(x+7)1/2]2b\lim_{b \to \infty} \left[-6(x+7)^{-1/2}\right]_2^b

STEP 10

**Plugging in** the bounds of integration: limb[6(b+7)1/2(6(2+7)1/2)]\lim_{b \to \infty} \left[-6(b+7)^{-1/2} - (-6(2+7)^{-1/2})\right]

STEP 11

**Simplifying**: limb[6(b+7)1/2+6(9)1/2]\lim_{b \to \infty} \left[-6(b+7)^{-1/2} + 6(9)^{-1/2}\right]

STEP 12

**Evaluating** the limit as bb approaches infinity: limb6(b+7)1/2=0\lim_{b \to \infty} -6(b+7)^{-1/2} = 0 Since the term (b+7)1/2(b+7)^{-1/2} goes to zero as bb goes to infinity.

STEP 13

And (9)1/2(9)^{-1/2} is just 13\frac{1}{3}, so the second term becomes: 613=26 \cdot \frac{1}{3} = 2

STEP 14

**Combining** the results: 0+2=20 + 2 = 2

STEP 15

The integral converges, and its value is **2**.

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