∫∞68(x+7)3/2dx

We are tasked with determining whether the following improper integral is divergent or convergent, and if convergent, we will evaluate it:

$\int_{6}^{\infty} \frac{1}{(x + 7)^{3/2}} \, dx$

Step 1: Determine Convergence or Divergence

The integral is improper because the upper limit is infinity. To analyze the behavior of the integral, let's start by checking whether the integrand behaves properly at infinity.

We need to examine the convergence of:

$\int_{6}^{\infty} \frac{1}{(x + 7)^{3/2}} \, dx$

Analyze the integrand at infinity:

For large values of $x$, the term $(x + 7)^{3/2}$ behaves like $x^{3/2}$. So, asymptotically, the integrand behaves like:

$\frac{1}{x^{3/2}}$

To determine if this integral converges, we compare it to the standard p-integral:

$\int_{a}^{\infty} \frac{1}{x^p} \, dx$

This integral converges if $p > 1$. In our case, $p = \frac{3}{2}$, which is greater than 1, so the integral converges.

Step 2: Evaluate the Integral

Now, let's proceed with the evaluation of the integral:

$\int_{6}^{\infty} \frac{1}{(x + 7)^{3/2}} \, dx$

We will make a substitution to simplify the integral.

Let:

$u = x + 7 \quad \Rightarrow \quad du = dx$

When $x = 6$, $u = 13$. When $x \to \infty$, $u \to \infty$.

Thus, the integral becomes:

$\int_{13}^{\infty} \frac{1}{u^{3/2}} \, du$

Now, evaluate this integral:

$\int_{13}^{\infty} u^{-3/2} \, du$

The antiderivative of $u^{-3/2}$ is:

$\frac{u^{-1/2}}{-1/2} = -2u^{-1/2} = -\frac{2}{\sqrt{u}}$

Now, apply the limits:

$\left[ -\frac{2}{\sqrt{u}} \right]_{13}^{\infty}$

At $u = \infty$, $\frac{2}{\sqrt{u}} \to 0$. At $u = 13$, we have:

$-\frac{2}{\sqrt{13}}$

Thus, the value of the integral is:

$0 - \left( -\frac{2}{\sqrt{13}} \right) = \frac{2}{\sqrt{13}}$

Conclusion

The integral is convergent, and the evaluated result is:

$\frac{2}{\sqrt{13}}$

---

Do you want more details or have any questions? Here are some additional related questions for practice:

1. Evaluate $\int_1^\infty \frac{1}{(x+2)^2} dx$.
2. Determine if $\int_2^\infty \frac{1}{x^3} dx$ is convergent or divergent.
3. Find the value of $\int_3^\infty \frac{1}{x^{5/2}} dx$.
4. Is $\int_0^\infty \frac{dx}{(x+1)^3}$ convergent or divergent?
5. Evaluate $\int_0^\infty \frac{1}{x^{3/2}} dx$.

Tip: When solving improper integrals, checking the behavior at infinity using the $p$-integral test is very useful!