Question

77–86. Comparison Test Determine whether the following integrals converge or diverge.84. ∫(from 1 to ∞) (2 + cos x) / x² dx

Accepted Answer

Identify the integral to analyze: $$\int_{1}^{\infty} \frac{2 + \cos x}{x$$^{2}$$} \, dx$$ and note that it is an improper integral because the upper limit is infinity. Recall the Comparison Test for improper integrals: if $$0 \leq f(x) \leq g(x)$$ for all $$x in$$ $$[1, \infty)$$ and $$\int_{1}^{\infty} g(x) \, dx$$ converges, then $$\int_{1}^{\infty} f(x) \, dx$$ also converges. Find a suitable function $$g(x) to $$compare with $$f(x) = \frac{2 + \cos x}{x$$^{2}$$}. $$Since $$\cos x$$ oscillates between $$-1$$ and $$1$$, the numerator $$2 + \cos x is $$bounded between $$1$$ and $$3. $$Therefore, $$\frac{2 + \cos x}{x$$^{2}$$} \leq \frac{3}{x$$^{2}$$}$$ for all $$x \geq 1. $$Check the convergence of the comparison integral $$\int_{1}^{\infty} \frac{3}{x$$^{2}$$} \, dx. $$Since $$\int_{1}^{\infty} \frac{1}{x$$^{2}$$} \, dx$$ converges (p-integral with $$p=2 \u003e 1$$), multiplying by a constant 3 does not affect convergence. Conclude by the Comparison Test that since $$\int_{1}^{\infty} \frac{3}{x$$^{2}$$} \, dx$$ converges and $$\frac{2 + \cos x}{x$$^{2}$$} \leq \frac{3}{x$$^{2}$$}$$, the original integral $$\int_{1}^{\infty} \frac{2 + \cos x}{x$$^{2}$$} \, dx$$ also converges.

77–86. Comparison Test Determine whether the following integrals converge or diverge.
84. ∫(from 1 to ∞) (2 + cos x) / x² dx

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주요 개념

Improper Integrals

Comparison Test for Improper Integrals

Behavior of Trigonometric Functions in Integrals