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Problem 285

# Determine if the following improper integrals of the first kind converge or diverge: a) $${ }^{\infty} \int_{1}\left(1 / \mathrm{x}^{\mathrm{p}}\right) \mathrm{dx}$$ b) $${ }^{\infty} \int_{0} \mathrm{e}^{-\mathrm{rx}} \mathrm{dx}$$ c) $$^{\infty} \int_{0} \sin x d x$$.

Expert verified
The improper integrals converge or diverge as follows: a) Converges for $$p > 1$$ and diverges for $$p \leq 1$$. b) Converges, and the value is $$\frac{1}{r}$$. c) Diverges.
See the step by step solution

## Step 1: Identify the Test

For improper integrals of functions of the form $$1 / x^p$$, we can apply the p-integral convergence test. In this case, our $$p$$ is given by the exponent of $$x$$.

## Step 2: Apply the p-integral Convergence Test

We compare our $$p$$ value to 1. The integral converges if $$p > 1$$, and diverges if $$p \leq 1$$. In this case, we are not given a specific value for $$p$$, so we cannot definitively conclude convergence or divergence.

## Step 3: Conclusion

The convergence of the integral depends on the value of $$p$$. The integral converges for $$p > 1$$ and diverges for $$p \leq 1$$. b) $${ }^{\infty} \int_{0} \mathrm{e}^{-\mathrm{rx}} \mathrm{dx}$$

## Step 1: Identify the Test

For improper integrals of functions involving exponentials, we can apply the exponential convergence test.

## Step 2: Apply the Exponential Convergence Test

To evaluate the improper integral, we can rewrite it as a limit: $\lim_{b \to \infty} \int_0^b e^{-rx} dx$ Now, we evaluate the integral: $\lim_{b \to \infty} \left[ -\frac{e^{-rx}}{r} \right]_0^b = \lim_{b \to \infty} \left(-\frac{e^{-rb}}{r} + \frac{1}{r}\right)$ Since $$r > 0$$, $$b \to \infty$$, we have $$e^{-rb} \to 0$$, so the limit is finite and equal to $\frac{1}{r}$

## Step 3: Conclusion

The improper integral converges, and the value of the integral is $$\frac{1}{r}$$. c) $$^{\infty} \int_{0} \sin x d x$$

## Step 1: Identify the Test

For improper integrals of trigonometric functions, we can apply the limit test or the comparison test. In this particular case, the limit test will work just fine.

## Step 2: Apply the Limit Test

Rewrite the improper integral as a limit: $\lim_{b \to \infty} \int_0^b \sin x dx$ Now, evaluate the integral: $\lim_{b \to \infty} \left[-\cos x \right]_0^b = \lim_{b \to \infty} (-\cos b + \cos 0)$ The limit does not exist because $$\cos b$$ oscillates between -1 and 1 as $$b$$ goes to infinity.

## Step 3: Conclusion

The improper integral diverges, since the limit of the integral does not exist.

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