For this question. I'm confused why the answer is pi/4. I circled in green where I was trying to work it out at the end. I understand partial fraction decomp , just not understanding the end part how we got pi/4. I thought answer was infinity because infinity - a number is infinity. ∫₀∞ (1/(x + 1)√(x² + 1)) dx = lim (b → ∞) [ (1/2) ln|x + 1| - (1/2) ln(x² + 1) + (1/2) tan⁻¹x ]₀ᵇ = lim (b → ∞) [ (1/2) ln( (b + 1) / √(b² + 1) ) + (1/2) tan⁻¹b ] - [(1/2) ln(1/√1) + (1/2) tan⁻¹0] = (1/2) ln1 + (1/2) - (1/2) ln1 - (1/2) x 0 = π/43. Write out the first five terms of the sequence... **Transcription of Mathematical Notes for Educational Website**---Given problem:\[ \int \frac{1}{x^2+1} \, dx \]Substitution method:1. **Substitution 1:** - Set \( u = x^2 + 1 \) - Then, \( du = 2x \, dx \)2. **Substitution 2:** - Set \( u = x^2 \) - Then, \( du = 2x \, dx \)---**Integration Steps:**\[ \int \frac{1}{2} \frac{1}{x^2+1} \, dx = \frac{1}{2} \left(\frac{-1}{x^2+1}\right) \left(-1\right) \]\[ = \frac{-1}{2} \left( - \frac{1}{x^2+1} \right) \]\[ = \frac{1}{2} \ln |x^2+1| + \text{const} \]\[ = \frac{1}{2} \ln|x^2+1| + \tan^{-1}x \]---**Evaluating the Limits from \(-\infty\) to \(+\infty\):**---(Calculation is within a green boxed region)1. Limit as \( x \to +\infty \):\[ \int_{0}^{+\infty} \frac{1}{x^2+1} \, dx = \frac{1}{2} \ln | +\infty + 1| \]\[ = \frac{1}{2} \ln \infty = \text{undefined} \]2. Limit as \( x \to 0 \):\[ \int_{0}^{0} \frac{1}{x^2+1} \, dx = 0 \]3. Limit as \( x \to -\infty \):\[ \int_{0}^{-\infty} \frac{1}{x^2+1} \, dx = \frac{1}{2} \ln | -\infty + 1 | \]\[ = \frac{1}{2} \ln \infty = \text{undefined} \]Computation yields results which suggest careful handling of limits,

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Answered: lim In ix 1 lim 를 In (는) + tan Ih | In…

lim In ix 1 lim 를 In (는) + tan Ih | In (x 1) + tan- x-, lim In () + tan-x %3D tan 3. Write out the first five terms of the

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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