×ジ* x* +x2dx500x 3+ 20ox+10000x

Here, we need to evaluate ∫2∞ x3+x2+x500x3+200x2+10000x. Let us first find…

Answered: xジャx*+x dx 500x 3+ 20ox+10000x

Calculus: Early Transcendentals

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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Step 1

Here, we need to evaluate $\int_{2}^{\infty} \frac{x^{3} + x^{2} + x}{500 x^{3} + 200 x^{2} + 10000 x}$ .

Let us first find $\int \frac{x^{3} + x^{2} + x}{500 x^{3} + 200 x^{2} + 10000 x}$ .

So,

$\int \frac{x^{3} + x^{2} + x}{500 x^{3} + 200 x^{2} + 10000 x} H e r e, \frac{x^{3} + x^{2} + x}{500 x^{3} + 200 x^{2} + 10000 x} = \frac{1}{500} + \frac{3 x - 95}{500 (5 x^{2} + 2 x + 100)} = \int \frac{1}{500} + \frac{3 x - 95}{500 (5 x^{2} + 2 x + 100)} d x = \int \frac{1}{500} d x + \int \frac{3 x - 95}{500 (5 x^{2} + 2 x + 100)} d x = \frac{x}{500} + \int \frac{3 x - 95}{500 (5 x^{2} + 2 x + 100)} d x = \frac{x}{500} + \frac{1}{500} \cdot \int \frac{3 x - 95}{5 x^{2} + 2 x + 100} d x = \frac{x}{500} + \frac{1}{500} (\int \frac{3 x}{5 x^{2} + 2 x + 100} d x - \int \frac{95}{5 x^{2} + 2 x + 100} d x) = \frac{x}{500} + \frac{1}{500} (\begin{matrix} 3 (\frac{1}{10} \ln |25 x^{2} + 10 x + 500| - \frac{1}{5 \sqrt{499}} \arctan (\frac{5 x + 1}{\sqrt{499}})) \\ - \frac{95}{\sqrt{499}} \arctan (\frac{5 x + 1}{\sqrt{499}}) \end{matrix}) = \frac{x}{500} + \frac{1}{5000 \sqrt{499}} (3 \sqrt{499} \ln |25 x^{2} + 10 x + 500| - 956 \arctan (\frac{5 x + 1}{\sqrt{499}})) = \frac{1}{500} x + \frac{3}{5000} \ln |25 x^{2} + 10 x + 500| - \frac{239}{1250 \sqrt{499}} \arctan (\frac{1}{\sqrt{499}} (5 x + 1))$