Question

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from π to ∞ of ((1 + sin x) / x² dx)

Accepted Answer

Identify the integral to be tested for convergence: $$\displaystyle \int_{\pi}^{\infty} \frac{1 + \sin x}{x$$^{2}$$} \, dx. $$Note that the integrand is $$\frac{1 + \sin x}{x$$^{2}$$}. $$Since $$\sin x$$ oscillates between $$-1$$ and $$1$$, the numerator $$1 + \sin x$$ oscillates between $$0$$ and $$2. To $$apply the Direct Comparison Test, find a simpler function that bounds the integrand from above and below. Since $$0 \leq 1 + \sin x \leq 2$$, we have $$0 \leq \frac{1 + \sin x}{x$$^{2}$$} \leq \frac{2}{x$$^{2}$$}$$ for $$x \geq \pi. $$Recall that the integral $$\int_{\pi}^{\infty} \frac{1}{x$$^{2}$$} \, dx$$ converges because it is a p-integral with $$p=2 \u003e 1. $$Therefore, by the Direct Comparison Test, since $$\frac{1 + \sin x}{x$$^{2}$$} \leq \frac{2}{x$$^{2}$$}$$ and the integral of $$\frac{2}{x$$^{2}$$}$$ converges, the original integral converges. Conclude that the integral $$\int_{\pi}^{\infty} \frac{1 + \sin x}{x$$^{2}$$} \, dx$$ converges by the Direct Comparison Test.

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from π to ∞ of ((1 + sin x) / x² dx)

Verified video answer for a similar problem:

Key Concepts

Improper Integrals and Convergence

Direct Comparison Test

Limit Comparison Test

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.∫ from π to ∞ of ((1 + sin x) / x² dx)

Verified video answer for a similar problem:

Key Concepts

Improper Integrals and Convergence

Direct Comparison Test

Limit Comparison Test

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from π to ∞ of ((1 + sin x) / x² dx)