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Use integration by parts to evaluate the following integral. \( \int_{-\infty}^{-6} \theta e^{\theta} d \theta \) Use the integration by parts formula to rewrite the given integral. Because the given integral is improper, rewrite it using a limit. A. \( \lim _{a \rightarrow \infty}-\frac{6}{e^{6}}-a e^{a}-\int_{-\infty}^{a} e^{\theta} d \theta \) B. \( \lim _{a \rightarrow-\infty}-\frac{6}{e^{6}}-a e^{a}-\int_{a}^{-6} e^{\theta} d \theta \) C. \( \lim _{a \rightarrow-\infty}-\frac{6}{e^{6}}-a e^{a}-\int_{a}^{\infty} e^{\theta} d \theta \) O D. \( \lim _{a \rightarrow \infty}-\frac{6}{e^{5}}-a e^{a}-\int_{a}^{-6} e^{\theta} d \theta \)

Ask by Clarke Ramos. in the United States
Mar 15,2025

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The correct answer is option B: \( \lim_{a \rightarrow-\infty}-\frac{6}{e^{6}}-a e^{a}-\int_{a}^{-6} e^{\theta} d \theta \).

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To solve the integral \( \int_{-\infty}^{-6} \theta e^{\theta} d \theta \) using integration by parts, we start by letting \( u = \theta \) and \( dv = e^{\theta} d\theta \). Thus, \( du = d\theta \) and \( v = e^{\theta} \). By applying the integration by parts formula, we also need to address the improper integral behavior at negative infinity while rewriting it with limits for evaluation. Now, let's rewrite the integral using limits: The correct formulation based on the properties of improper integrals would be: B. \( \lim _{a \rightarrow -\infty}-\frac{6}{e^{6}}-a e^{a}-\int_{a}^{-6} e^{\theta} d \theta \) This choice correctly sets the limit as \( a \) approaches negative infinity and accounts for the correct bounds for the integral. On a historical note, integration by parts is rooted in the work of mathematicians like Gottfried Wilhelm Leibniz in the 17th century who contributed significantly to calculus. Integration techniques have evolved to become essential tools in mathematics, enabling the evaluation of complex integrals that appear in various scientific fields. For real-world applications, integration by parts comes in handy not just in pure mathematics but in fields such as physics and engineering. Whether calculating the area under a curve or solving problems involving exponential growth and decay, this integration technique helps simplify complex problems into manageable pieces, akin to taking apart a puzzle to reveal the picture within.

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