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# A3 - f x ≥ g x ≥ for x ≥ a • If R ∞ a f x dx is...

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Math 128: Assignment 3 due Wed, Oct. 5 by 11:59 pm Koray Karabina ([email protected]) 1. Read “Section 7.5 Strategy for Integration” in the course textbook. 2. Determine whether each integral is convergent or divergent. Evaluate those that are convergent. (a) R 1 x x 2 +4 dx (b) R 0 x e x dx (c) R -∞ cos ( πx ) dx (d) R -∞ x - 1 x 3 dx 3. Determine whether each integral is convergent or divergent. Evaluate those that are convergent. (a) R π/ 2 π/ 4 sin x cos x dx (b) R 1 0 ln x x dx (c) R 1 0 1 4 x - 1 dx (d) R 1 0 1 1 - x 2 dx 4. The purpose of this question is to teach how to apply the Comparison Theorem to determine whether an improper integral is convergent or divergent. The statement of the Comparison Theorem is as follows (we will omit the proof for now): Theorem 1 (Comparison Theorem) . Suppose that f and g are continuous functions with
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Unformatted text preview: f ( x ) ≥ g ( x ) ≥ for x ≥ a . • If R ∞ a f ( x ) dx is convergent then R ∞ a g ( x ) dx is convergent. • If R ∞ a g ( x ) dx is divergent then R ∞ a f ( x ) dx is divergent. Use the Comparison Theorem to determine whether the integral is convergent or divergent. You do not have to evaluate the integrals. You might have a look at Example 9 and Example 10 in Section 7.8 in the course textbook. (a) R ∞ 1 2+ e-x x dx Hint: Try to ﬁnd some relation between 2+ e-x x and 1 x for x ≥ 1. (b) R ∞ ( x 3 e-2 x ) dx Hint: Show that x 3 e-2 x ≤ e-x for suﬃciently large x . You might use the fact that x ≥ 3 ln x for all x ≥ 5....
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